gdZddlmZmZddlmZddlmZddlmZddl m Z ddl m Z dd l mZdd lmZd ZGd d ZeZdZeddddZdedefdZd"dZdZd#dZdZd$dZd%dZd Zd!ZdS)&z" Generating and counting primes. )bisect bisect_leftcount)array)randint)sqrt)isprime) deprecated)as_intc>ddlm}t||S)z Wrapping ceiling in as_int will raise an error if there was a problem determining whether the expression was exactly an integer or not.r)ceiling)#sympy.functions.elementary.integersrr )ars R/var/www/html/ai-engine/env/lib/python3.11/site-packages/sympy/ntheory/generate.py_as_int_ceilingrs,<;;;;; ''!**  cfeZdZdZddZdZddZdZdZd Z dd Z d Z d Z d Z dZdZdZdS)SieveaA list of prime numbers, implemented as a dynamically growing sieve of Eratosthenes. When a lookup is requested involving an odd number that has not been sieved, the sieve is automatically extended up to that number. Implementation details limit the number of primes to ``2^32-1``. Examples ======== >>> from sympy import sieve >>> sieve._reset() # this line for doctest only >>> 25 in sieve False >>> sieve._list array('L', [2, 3, 5, 7, 11, 13, 17, 19, 23]) @Bc6d_tdgd_tdgd_tdgd_|dkrt d|_tfd jjjfDsJd S) z Initial parameters for the Sieve class. Parameters ========== sieve_interval (int): Amount of memory to be used Raises ====== ValueError If ``sieve_interval`` is not positive. L) )rr r rri)rr r#rr#rz+sieve_interval should be a positive integerc3HK|]}t|jkVdSN)len_n).0r"selfs r z!Sieve.__init__..Cs0UU3q66TW$UUUUUUrN)r'_array_list_tlist_mlist ValueErrorsieve_intervalall)r)r0s` r__init__zSieve.__init__-sC!5!5!566 S"4"4"455 S"7"7"788 Q  JKK K,UUUUtz4; .TUUUUUUUUUUrcddt|j|jd|jd|jd|jd|jddt|j|jd|jd|jd|jd|jdd t|j|jd|jd|jd|jd|jdfzS) Nzs<%s sieve (%i): %i, %i, %i, ... %i, %i %s sieve (%i): %i, %i, %i, ... %i, %i %s sieve (%i): %i, %i, %i, ... %i, %i>primerr rr#totientmobius)r&r,r-r.)r)s r__repr__zSieve.__repr__Es6c$*ooA 1 tz!}BB DK((QQQR$+b/ s4;''QQQR$+b/ :C C CrNctd|||fDrdx}x}}|r|jd|j|_|r|jd|j|_|r|jd|j|_dSdS)z]Reset all caches (default). To reset one or more set the desired keyword to True.c3K|]}|duV dSr%)r(r"s rr*zSieve._reset..Vs&;;QqDy;;;;;;rTN)r1r,r'r-r.)r)r4r6r7s r_resetz Sieve._resetSs ;;5'6":;;; ; ; ,'+ +E +Gf  .HTWH-DJ  0+htwh/DK  0+htwh/DKKK 0 0rc Rt|}|jddz}||krdS|dz}||kr?|xjtd|||z c_||dz}}||k?|xjtd|||dzz c_dS)zGrow the sieve to cover all primes <= n. Examples ======== >>> from sympy import sieve >>> sieve._reset() # this line for doctest only >>> sieve.extend(30) >>> sieve[10] == 29 True r#r Nrr)intr,r+ _primerange)r)nnumnum2s rextendz Sieve.extend_s FFjnq  s77 FAvaii JJ&d&6&6sD&A&ABB BJJdAgCaii fS$"2"23A">">??? rc #K|dzr|dz}||krt|j||z dz}dg|z}|jdt|jt |d|zzdzD](}t |dz|z dz|z||D]}d||<)t |D]\}}|r |d|zzdzV|d|zz }||kdSdS)a? Generate all prime numbers in the range (a, b). Parameters ========== a, b : positive integers assuming the following conditions * a is an even number * 2 < self._list[-1] < a < b < nextprime(self._list[-1])**2 Yields ====== p (int): prime numbers such that ``a < p < b`` Examples ======== >>> from sympy.ntheory.generate import Sieve >>> s = Sieve() >>> s._list[-1] 13 >>> list(s._primerange(18, 31)) [19, 23, 29] rr TFN)minr0r,rr range enumerate)r)rb block_sizeblockptidxs rr?zSieve._primerangexs'4 q5  FA!eeT01q5Q,??JFZ'EZ&T!a*n:Lq:P5Q5Q"R"R RS % %!a%!)  1Q6 AFF%%A$E!HH%#E** * *Q*a#g+/))) Z A!eeeeeerct|}t|j|krJ|t |jddzt|j|kHdSdS)aExtend to include the ith prime number. Parameters ========== i : integer Examples ======== >>> from sympy import sieve >>> sieve._reset() # this line for doctest only >>> sieve.extend_to_no(9) >>> sieve._list array('L', [2, 3, 5, 7, 11, 13, 17, 19, 23]) Notes ===== The list is extended by 50% if it is too short, so it is likely that it will be longer than requested. r#g?N)r r&r,rCr>)r)r"s r extend_to_nozSieve.extend_to_nosh. 1II$*oo!! KKDJrNS011 2 2 2$*oo!!!!!!rc#:K|t|}d}n,tdt|}t|}||krdS|||jt |j|t |j|Ed{VdS)a(Generate all prime numbers in the range [2, a) or [a, b). Examples ======== >>> from sympy import sieve, prime All primes less than 19: >>> print([i for i in sieve.primerange(19)]) [2, 3, 5, 7, 11, 13, 17] All primes greater than or equal to 7 and less than 19: >>> print([i for i in sieve.primerange(7, 19)]) [7, 11, 13, 17] All primes through the 10th prime >>> list(sieve.primerange(prime(10) + 1)) [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] Nr)rmaxrCr,r)r)rrHs r primerangezSieve.primeranges0 9""AAAAq))**A""A 66 F A:k$*a88)$*a889: : : : : : : : : :rc #Ktdt|}t|}t|j}||krdS||kr$t ||D]}|j|VdS|xjt dt ||z c_t d|D]g}|j|}||dz krE||zdz |z|z}t |||D]%}|j|xx|j||zzcc<&||kr|Vht ||D]a}|j|}||kr7t |||D]%}|j|xx|j||zzcc<&||kr|j|VbdS)zGenerate all totient numbers for the range [a, b). Examples ======== >>> from sympy import sieve >>> print([i for i in sieve.totientrange(7, 18)]) [6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16] r Nr)rQrr&r-rFr+)r)rrHr@r"ti startindexjs r totientrangezSieve.totientranges ?1%% & & A      66 F !VV1a[[ % %k!n$$$$ % % KK6#uQ{{33 3KK1a[[  [^Q;;"#a%!)!1A!5J":q!44>> A$+a.A*==66HHH1a[[ ) )[^77"1a^^>> A$+a.A*==66+a.(((  ) )rc#Ktdt|}t|}t|j}||krdS||kr$t ||D]}|j|VdS|xjt ddg||z zz c_t d|D]P}|j|}||zdz |z|z}t |||D]}|j|xx|zcc<||kr|VQt ||D]E}|j|}t d|z||D]}|j|xx|zcc<||kr|VFdS)aGenerate all mobius numbers for the range [a, b). Parameters ========== a : integer First number in range b : integer First number outside of range Examples ======== >>> from sympy import sieve >>> print([i for i in sieve.mobiusrange(7, 18)]) [-1, 0, 0, 1, -1, 0, -1, 1, 1, 0, -1] r Nr"rr)rQrr&r.rFr+)r)rrHr@r"mirUrVs r mobiusrangezSieve.mobiusranges& ?1%% & & A      66 F !VV1a[[ % %k!n$$$$ % % KK6#sAE{33 3KK1a[[  [^!eaiA-1 z1a00))AKNNNb(NNNN66HHH1a[[  [^q1ua++))AKNNNb(NNNN66HHH   rc"t|}t|}|dkrtd|z||jdkr||t |j|}|j|dz |kr||fS||dzfS)a~Return the indices i, j of the primes that bound n. If n is prime then i == j. Although n can be an expression, if ceiling cannot convert it to an integer then an n error will be raised. Examples ======== >>> from sympy import sieve >>> sieve.search(25) (9, 10) >>> sieve.search(23) (9, 9) rzn should be >= 2 but got: %sr#r )rr r/r,rCr)r)r@testrHs rsearchz Sieve.search1s"q!! 1II q55;a?@@ @ tz"~   KKNNN 4:q ! ! :a!e  $ $a4Ka!e8Orc t|}|dksJn#ttf$rYdSwxYw|dzdkr|dkS||\}}||kS)NrFr)r r/AssertionErrorr])r)r@rrHs r __contains__zSieve.__contains__Nsz q A66666N+   55  q5A::6M{{1~~1Av s //c#BKtdD] }||V dS)Nr r)r)r@s r__iter__zSieve.__iter__Ys4q  Aq'MMMM  rc|t|tr_||j|j|jnd}|dkrt d|j|dz |jdz |jS|dkrt dt|}|||j|dz S)zReturn the nth prime numberNrr zSieve indices start at 1.) isinstanceslicerOstopstart IndexErrorr,stepr )r)r@rgs r __getitem__zSieve.__getitem__]s a   %   af % % % !w2AGGEqyy!!<===:eai 169: :1uu!!<===q A   a :a!e$ $r)r)NNNr%)__name__ __module__ __qualname____doc__r2r8r<rCr?rOrRrWrZr]r`rbrjr;rrrrs$VVVV0 C C C 0 0 0 0@@@2' ' ' R3336":":":":H#)#)#)J***X:   %%%%%rrc t|}|dkrtd|ttjkr t|Sddlm}ddlm}d}t||||||zz}||kr%||zdz }|||kr|}n|dz}||k%t|dz }||krt|r|dz }|dz }||k|dz S)aK Return the nth prime, with the primes indexed as prime(1) = 2, prime(2) = 3, etc.... The nth prime is approximately $n\log(n)$. Logarithmic integral of $x$ is a pretty nice approximation for number of primes $\le x$, i.e. li(x) ~ pi(x) In fact, for the numbers we are concerned about( x<1e11 ), li(x) - pi(x) < 50000 Also, li(x) > pi(x) can be safely assumed for the numbers which can be evaluated by this function. Here, we find the least integer m such that li(m) > n using binary search. Now pi(m-1) < li(m-1) <= n, We find pi(m - 1) using primepi function. Starting from m, we have to find n - pi(m-1) more primes. For the inputs this implementation can handle, we will have to test primality for at max about 10**5 numbers, to get our answer. Examples ======== >>> from sympy import prime >>> prime(10) 29 >>> prime(1) 2 >>> prime(100000) 1299709 See Also ======== sympy.ntheory.primetest.isprime : Test if n is prime primerange : Generate all primes in a given range primepi : Return the number of primes less than or equal to n References ========== .. [1] https://en.wikipedia.org/wiki/Prime_number_theorem#Table_of_.CF.80.28x.29.2C_x_.2F_log_x.2C_and_li.28x.29 .. [2] https://en.wikipedia.org/wiki/Prime_number_theorem#Approximations_for_the_nth_prime_number .. [3] https://en.wikipedia.org/wiki/Skewes%27_number r z-nth must be a positive integer; prime(1) == 2rloglir) r r/r&siever,&sympy.functions.elementary.exponentialrq'sympy.functions.special.error_functionsrsr>_primepir )nthr@rqrsrrHmidn_primess rr4r4vs2b s A1uuHIIIC  Qx:::::::::::: A Ass1vvCCFF # $%%A a%%1ul 2c77Q;;AAaA a%% AH Q,, 1::  MH Q Q,, q5LrzgThe `sympy.ntheory.generate.primepi` has been moved to `sympy.functions.combinatorial.numbers.primepi`.z1.13z%deprecated-ntheory-symbolic-functions)deprecated_since_versionactive_deprecations_targetc$ddlm}||S)a Represents the prime counting function pi(n) = the number of prime numbers less than or equal to n. .. deprecated:: 1.13 The ``primepi`` function is deprecated. Use :class:`sympy.functions.combinatorial.numbers.primepi` instead. See its documentation for more information. See :ref:`deprecated-ntheory-symbolic-functions` for details. Algorithm Description: In sieve method, we remove all multiples of prime p except p itself. Let phi(i,j) be the number of integers 2 <= k <= i which remain after sieving from primes less than or equal to j. Clearly, pi(n) = phi(n, sqrt(n)) If j is not a prime, phi(i,j) = phi(i, j - 1) if j is a prime, We remove all numbers(except j) whose smallest prime factor is j. Let $x= j \times a$ be such a number, where $2 \le a \le i / j$ Now, after sieving from primes $\le j - 1$, a must remain (because x, and hence a has no prime factor $\le j - 1$) Clearly, there are phi(i / j, j - 1) such a which remain on sieving from primes $\le j - 1$ Now, if a is a prime less than equal to j - 1, $x= j \times a$ has smallest prime factor = a, and has already been removed(by sieving from a). So, we do not need to remove it again. (Note: there will be pi(j - 1) such x) Thus, number of x, that will be removed are: phi(i / j, j - 1) - phi(j - 1, j - 1) (Note that pi(j - 1) = phi(j - 1, j - 1)) $\Rightarrow$ phi(i,j) = phi(i, j - 1) - phi(i / j, j - 1) + phi(j - 1, j - 1) So,following recursion is used and implemented as dp: phi(a, b) = phi(a, b - 1), if b is not a prime phi(a, b) = phi(a, b-1)-phi(a / b, b-1) + phi(b-1, b-1), if b is prime Clearly a is always of the form floor(n / k), which can take at most $2\sqrt{n}$ values. Two arrays arr1,arr2 are maintained arr1[i] = phi(i, j), arr2[i] = phi(n // i, j) Finally the answer is arr2[1] Examples ======== >>> from sympy import primepi, prime, prevprime, isprime >>> primepi(25) 9 So there are 9 primes less than or equal to 25. Is 25 prime? >>> isprime(25) False It is not. So the first prime less than 25 must be the 9th prime: >>> prevprime(25) == prime(9) True See Also ======== sympy.ntheory.primetest.isprime : Test if n is prime primerange : Generate all primes in a given range prime : Return the nth prime r)primepi)%sympy.functions.combinatorial.numbersr~)r@ func_primepis rr~r~s&pNMMMMM <??rr@returnc |dkrdS|tjdkr t|dSt|}dg|dzz}dg|dzz}t d|dzD]}|dz ||<||zdz ||<t d|dzD]}||||dz kr||dz }t dt |||zz|dzD]C}||z}||kr||xx|||z zcc<'||xx|||z|z zcc<Dt |||zdz }t ||dD]}||xx|||z|z zcc<|dS)a Represents the prime counting function pi(n) = the number of prime numbers less than or equal to n. Explanation =========== In sieve method, we remove all multiples of prime p except p itself. Let phi(i,j) be the number of integers 2 <= k <= i which remain after sieving from primes less than or equal to j. Clearly, pi(n) = phi(n, sqrt(n)) If j is not a prime, phi(i,j) = phi(i, j - 1) if j is a prime, We remove all numbers(except j) whose smallest prime factor is j. Let $x= j \times a$ be such a number, where $2 \le a \le i / j$ Now, after sieving from primes $\le j - 1$, a must remain (because x, and hence a has no prime factor $\le j - 1$) Clearly, there are phi(i / j, j - 1) such a which remain on sieving from primes $\le j - 1$ Now, if a is a prime less than equal to j - 1, $x= j \times a$ has smallest prime factor = a, and has already been removed(by sieving from a). So, we do not need to remove it again. (Note: there will be pi(j - 1) such x) Thus, number of x, that will be removed are: phi(i / j, j - 1) - phi(j - 1, j - 1) (Note that pi(j - 1) = phi(j - 1, j - 1)) $\Rightarrow$ phi(i,j) = phi(i, j - 1) - phi(i / j, j - 1) + phi(j - 1, j - 1) So,following recursion is used and implemented as dp: phi(a, b) = phi(a, b - 1), if b is not a prime phi(a, b) = phi(a, b-1)-phi(a / b, b-1) + phi(b-1, b-1), if b is prime Clearly a is always of the form floor(n / k), which can take at most $2\sqrt{n}$ values. Two arrays arr1,arr2 are maintained arr1[i] = phi(i, j), arr2[i] = phi(n // i, j) Finally the answer is arr2[1] Parameters ========== n : int rrr#r )rtr,r]r rFrE) r@limarr1arr2r"rKrVstlim2s rrwrwsx 1uuqEKO||Aq!! q''C 3#'?D 3#'?D 1cAg  a%Qq&1*Q 1cAg  (( 7d1q5k ! !  QKq#aAElC001455 - -AQBSyyQ48a<'Q4R=1,,3A ""sD"%% ( (A GGGtAF|a' 'GGGG ( 7Nrc.t|}t|}|dkrtd|dkrd}|dz}|tjdkrt|\}}||zdz t tjkrtj||zdz Sttjd||zt tjz Sd|kr#t|D]}t|}|Sd|dzz}||kr|dz }t|r|S|dz }n*||z d kr|dz }t|r|S|dz }n|d z} t|r|S|dz }t|r|S|dz }-) aU Return the ith prime greater than n. Parameters ========== n : integer ith : positive integer Returns ======= int : Return the ith prime greater than n Raises ====== ValueError If ``ith <= 0``. If ``n`` or ``ith`` is not an integer. Notes ===== Potential primes are located at 6*j +/- 1. This property is used during searching. >>> from sympy import nextprime >>> [(i, nextprime(i)) for i in range(10, 15)] [(10, 11), (11, 13), (12, 13), (13, 17), (14, 17)] >>> nextprime(2, ith=2) # the 2nd prime after 2 5 See Also ======== prevprime : Return the largest prime smaller than n primerange : Generate all primes in a given range rzith should be positiverr r5r#rr!r) r>r r/rtr,r]r& nextprimerFr )r@ithr"l_nns rrrusP AAs AAvv12221uu  QEKO||A1 q519s5;'' ' ';q1uqy) )R!a%#ek2B2B*BCCC1uuq  A! AA AqDB Qww Q 1:: H Q R1 Q 1:: H Q F 1:: H Q 1:: H Q rct|}|dkrtd|dkrdddddd|S|tjdkr@t|\}}||krt|dz St|Sd |d zz}||z dkr|dz }t |r|S|d z}n|dz} t |r|S|dz}t |r|S|d z}-) a Return the largest prime smaller than n. Notes ===== Potential primes are located at 6*j +/- 1. This property is used during searching. >>> from sympy import prevprime >>> [(i, prevprime(i)) for i in range(10, 15)] [(10, 7), (11, 7), (12, 11), (13, 11), (14, 13)] See Also ======== nextprime : Return the ith prime greater than n primerange : Generates all primes in a given range rzno preceding primesrr)rr!rrrr#r rr!)rr/rtr,r]r )r@rurs r prevprimers& A1uu.///1uuqQ1--a00EKO||A1 661: 8O AqDB2v{{ F 1:: H Q F 1:: H Q 1:: H Q rNc#K|d|}}||krdStjd}||kr#t||Ed{VdS||kr8tjttj|dEd{V|dz}n |dzr|dz}t ||dz}||kr#t||Ed{V|}||krdS t |}||kr|VndS)a Generate a list of all prime numbers in the range [2, a), or [a, b). If the range exists in the default sieve, the values will be returned from there; otherwise values will be returned but will not modify the sieve. Examples ======== >>> from sympy import primerange, prime All primes less than 19: >>> list(primerange(19)) [2, 3, 5, 7, 11, 13, 17] All primes greater than or equal to 7 and less than 19: >>> list(primerange(7, 19)) [7, 11, 13, 17] All primes through the 10th prime >>> list(primerange(prime(10) + 1)) [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] The Sieve method, primerange, is generally faster but it will occupy more memory as the sieve stores values. The default instance of Sieve, named sieve, can be used: >>> from sympy import sieve >>> list(sieve.primerange(1, 30)) [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] Notes ===== Some famous conjectures about the occurrence of primes in a given range are [1]: - Twin primes: though often not, the following will give 2 primes an infinite number of times: primerange(6*n - 1, 6*n + 2) - Legendre's: the following always yields at least one prime primerange(n**2, (n+1)**2+1) - Bertrand's (proven): there is always a prime in the range primerange(n, 2*n) - Brocard's: there are at least four primes in the range primerange(prime(n)**2, prime(n+1)**2) The average gap between primes is log(n) [2]; the gap between primes can be arbitrarily large since sequences of composite numbers are arbitrarily large, e.g. the numbers in the sequence n! + 2, n! + 3 ... n! + n are all composite. See Also ======== prime : Return the nth prime nextprime : Return the ith prime greater than n prevprime : Return the largest prime smaller than n randprime : Returns a random prime in a given range primorial : Returns the product of primes based on condition Sieve.primerange : return range from already computed primes or extend the sieve to contain the requested range. References ========== .. [1] https://en.wikipedia.org/wiki/Prime_number .. [2] https://primes.utm.edu/notes/gaps.html Nrr#r )rtr,rRrrEr?r)rrHlargest_known_primetails rrRrRsOV y!1Avv+b/ ##Aq))))))))) ;{5;::;;<<<<<<<< ! # Q Q q&* + +D4xx$$Q--------- Avv aLL q55GGGG F rc||krdStt||f\}}t|dz |}t|}||krt |}||krt d|S)aG Return a random prime number in the range [a, b). Bertrand's postulate assures that randprime(a, 2*a) will always succeed for a > 1. Note that due to implementation difficulties, the prime numbers chosen are not uniformly random. For example, there are two primes in the range [112, 128), ``113`` and ``127``, but ``randprime(112, 128)`` returns ``127`` with a probability of 15/17. Examples ======== >>> from sympy import randprime, isprime >>> randprime(1, 30) #doctest: +SKIP 13 >>> isprime(randprime(1, 30)) True See Also ======== primerange : Generate all primes in a given range References ========== .. [1] https://en.wikipedia.org/wiki/Bertrand's_postulate Nr z&no primes exist in the specified range)mapr>rrrr/)rrHr@rKs r randprimer[sy@ Avv sQF  DAqAqA! AAvv aLL1uuABBB HrTc|rt|}nt|}|dkrtdd}|r)td|dzD]}|t |z}nt d|dzD]}||z}|S)a: Returns the product of the first n primes (default) or the primes less than or equal to n (when ``nth=False``). Examples ======== >>> from sympy.ntheory.generate import primorial, primerange >>> from sympy import factorint, Mul, primefactors, sqrt >>> primorial(4) # the first 4 primes are 2, 3, 5, 7 210 >>> primorial(4, nth=False) # primes <= 4 are 2 and 3 6 >>> primorial(1) 2 >>> primorial(1, nth=False) 1 >>> primorial(sqrt(101), nth=False) 210 One can argue that the primes are infinite since if you take a set of primes and multiply them together (e.g. the primorial) and then add or subtract 1, the result cannot be divided by any of the original factors, hence either 1 or more new primes must divide this product of primes. In this case, the number itself is a new prime: >>> factorint(primorial(4) + 1) {211: 1} In this case two new primes are the factors: >>> factorint(primorial(4) - 1) {11: 1, 19: 1} Here, some primes smaller and larger than the primes multiplied together are obtained: >>> p = list(primerange(10, 20)) >>> sorted(set(primefactors(Mul(*p) + 1)).difference(set(p))) [2, 5, 31, 149] See Also ======== primerange : Generate all primes in a given range r zprimorial argument must be >= 1r)r r>r/rFr4rR)r@rxrKr"s r primorialrsd  1II FF1uu:;;; A q!a%  A qMAA Aq1u%%  A FAA HrFc#Kt|pd}dx}}|||}}d}|r|V||kr@|r||kr8|dz }||kr |}|dz}d}|r|V||}|dz }||kr|2||k8|r||kr |rdS|dfVdS|sRd} |x}}t|D] }||}||kr!||}||}| dz } ||k!|| fVdSdS)awFor a given iterated sequence, return a generator that gives the length of the iterated cycle (lambda) and the length of terms before the cycle begins (mu); if ``values`` is True then the terms of the sequence will be returned instead. The sequence is started with value ``x0``. Note: more than the first lambda + mu terms may be returned and this is the cost of cycle detection with Brent's method; there are, however, generally less terms calculated than would have been calculated if the proper ending point were determined, e.g. by using Floyd's method. >>> from sympy.ntheory.generate import cycle_length This will yield successive values of i <-- func(i): >>> def gen(func, i): ... while 1: ... yield i ... i = func(i) ... A function is defined: >>> func = lambda i: (i**2 + 1) % 51 and given a seed of 4 and the mu and lambda terms calculated: >>> next(cycle_length(func, 4)) (6, 3) We can see what is meant by looking at the output: >>> iter = cycle_length(func, 4, values=True) >>> list(iter) [4, 17, 35, 2, 5, 26, 14, 44, 50, 2, 5, 26, 14] There are 6 repeating values after the first 3. If a sequence is suspected of being longer than you might wish, ``nmax`` can be used to exit early (and mu will be returned as None): >>> next(cycle_length(func, 4, nmax = 4)) (4, None) >>> list(cycle_length(func, 4, nmax = 4, values=True)) [4, 17, 35, 2] Code modified from: https://en.wikipedia.org/wiki/Cycle_detection. rr rN)r>rF) fx0nmaxvaluespowerlamtortoiseharer"mus r cycle_lengthrsf tyq>>DOEC2dH A  d  D AHH Q C<<H QJEC  JJJqww q d  D AHH T   F*    F   4s  A1T77DD$q{{H1T77D !GB$2g   rc t|}|dkrtdgd}|dkr ||dz Sdtjd}}||t |z dz krN||dz kr/||zdz }|t |z dz |kr|}n|}||dz k/t |r|dz}|Sddlm}dd lm }d}t||||||zz}||kr+||zdz }|||z dz |kr|}n|dz}||k+|t |z dz }||krt |s|dz}|dz}||kt |r|dz}|S) a Return the nth composite number, with the composite numbers indexed as composite(1) = 4, composite(2) = 6, etc.... Examples ======== >>> from sympy import composite >>> composite(36) 52 >>> composite(1) 4 >>> composite(17737) 20000 See Also ======== sympy.ntheory.primetest.isprime : Test if n is prime primerange : Generate all primes in a given range primepi : Return the number of primes less than or equal to n prime : Return the nth prime compositepi : Return the number of positive composite numbers less than or equal to n r z1nth must be a positive integer; composite(1) == 4) r!rr rr!r#rrprr) r r/rtr,rwr rurqrvrsr>) rxr@ composite_arrrrHryrqrs n_compositess r compositer!s0 s A1uuLMMM888MBwwQU## ek"oqAA Oa !a%iiq5Q,CXc]]"Q&** !a%ii 1::  FA:::::::::::: A Ass1vvCCFF # $%%A a%%1ul C=1 q AAaA a%%x{{?Q&L   qzz  A L Q   qzz Q HrcZt|}|dkrdS|t|z dz S)ak Return the number of positive composite numbers less than or equal to n. The first positive composite is 4, i.e. compositepi(4) = 1. Examples ======== >>> from sympy import compositepi >>> compositepi(25) 15 >>> compositepi(1000) 831 See Also ======== sympy.ntheory.primetest.isprime : Test if n is prime primerange : Generate all primes in a given range prime : Return the nth prime primepi : Return the number of primes less than or equal to n composite : Return the nth composite number r!rr )r>rw)r@s r compositepirbs2, AA1uuq x{{?Q r)r r%)T)NF) rnrr itertoolsrrr+sympy.core.randomrsympy.external.gmpyr primetestr sympy.utilities.decoratorr sympy.utilities.miscr rrrtr4r~r>rwrrrRrrrrrr;rrrs '&&&&&&&"!!!!!%%%%%%$$$$$$000000''''''V%V%V%V%V%V%V%V%r IIIV  kBDDDUU DDUpUsUsUUUUpKKKK\,,,^ffffR) ) ) X? ? ? ? DUUUUp> > > Br