a RG5dUã@s¤ddlmZmZmZmZmZmZmZmZm Z m Z m Z m Z m Z mZmZmZmZddlmZddlmZmZmZmZddlZdd„Zdd „Zd d „Zdd d„ZdgZdS)é)ÚfzeroÚfrom_intÚ from_rationalÚfoneÚfhalfÚbitcountÚto_intÚto_strÚmpf_mulÚmpf_divÚmpf_subÚmpf_addÚmpf_sqrtÚmpf_piÚ mpf_cosh_sinhÚmpf_cosÚmpf_sin)Úigcdé)Ú_sqrt_mod_prime_powerÚlegendre_symbolÚ jacobi_symbolÚis_quad_residueNcCsæd}dg|adg|at|dƒd}td|ƒD]:}t|dkr2t||||ƒD]}t|dkrR|t|<qRq2td|ƒD]h}t|dkrž|t|<|dt|<qxt|}||}||dkrÌt||t|<qxt||dt|<qxdS)Né †rrçà?éé)Ú_factorÚ_totientÚintÚrange)ÚmaxnÚlimÚiÚjÚxÚy©r'úU/var/www/html/django/DPS/env/lib/python3.9/site-packages/sympy/ntheory/partitions_.pyÚ_pre s&        r)cCsR|dkr tS|}d}t|}||dkr:||}|d7}q||}dd|}t|ƒ}|dkr |dkrd|} | || }|td|d| ƒ| }t|d|dƒd} tttd| ƒ||ƒt| ƒ|ƒ} tttd |t| d| ƒƒt t|ƒ|ƒ|ƒt | |ƒ|ƒS|dkr¾d|} | || }|dkrH|td |dd| ƒ| }t|d|dƒd} tttd| ƒ||ƒt| ƒ|ƒ} tttdd |dt | dƒƒt t|dƒ|ƒ|ƒt | |ƒ|ƒS|||}||dkr|dkrtttd|ƒƒt t|ƒ|ƒ|ƒSt St ||ƒst S||d|d} |td | d|ƒ}t|||ƒd} tttd| ƒ||ƒt|ƒ|ƒ} tttdtd|ƒƒt t|ƒ|ƒ|ƒt| |ƒ|ƒS|dks´|dkrrt|dƒt|dƒ} }d| |}||||dd| t||||t|d|ƒ|}| |||dd|t|||| t|d|ƒ|}tt|||ƒt|||ƒ|ƒS|dkròd|d td t|d|ƒ|}d|d|ddd|ddd}tttd ƒt|||ƒ|ƒt|||ƒƒSd|dtdt|d|ƒ|}d||ddddd}tt|||ƒt|||ƒ|ƒS)z‹ Compute the inner sum in HRR formula [1]_ References ========== .. [1] http://msp.org/pjm/1956/6-1/pjm-v6-n1-p18-p.pdf rréréé éééÿÿÿÿé@i@ré€é )rrrÚpowrr r rrrrrrrrrrÚ_a)ÚnÚkÚprecÚk1ÚeÚpÚk2ÚvÚpiÚmodÚmÚargZ_phiÚd1Úd2Ún1Ún2r'r'r(r4"s¸       ÿÿþ ý   ÿþ ý   þ þþ ý ÿÿÿÿ $, þ ý$ r4c Cs”t|ƒ}t|ƒ}t|||ƒ}tt|ƒtdd|ƒ|ƒ}t||ƒ}tt||ƒ|ƒ\} } tt||ƒtt||ƒ|ƒ|ƒ} tt|| ƒt| ||ƒ|ƒ} t| | ƒS)z„ Compute the sinh term in the outer sum of the HRR formula. The constants sqrt(2/3*pi) and sqrt(8) must be precomputed. rr*)rrr r rrrr ) r5r$r7Úsq23piÚsqrt8r=ÚaÚbÚcÚchÚshÚDÚEr'r'r(Ú_dxs  þrNFc Csrt|ƒ}|dkrdS|dkr(gd¢|Sdtƒvr8tƒttjd|ddt d|¡t d ¡d ƒt d d¡}t|d d ƒ}}t}td td|ddƒƒ}|dkr¼tdƒ‚t t t dd|ƒ|ƒt |ƒ|ƒ}t t dƒ|ƒ}td |ƒD]j} t|| |ƒ} t|| |||ƒ} t|t | | ƒ|ƒ}|rHtd| d|t| d ƒt| d ƒƒttt| ƒƒƒd}qòttt|t|ƒƒƒS)aæ Calculate the partition function P(n), i.e. the number of ways that n can be written as a sum of positive integers. P(n) is computed using the Hardy-Ramanujan-Rademacher formula [1]_. The correctness of this implementation has been tested through $10^{10}$. Examples ======== >>> from sympy.ntheory import npartitions >>> npartitions(25) 1958 References ========== .. [1] http://mathworld.wolfram.com/PartitionFunctionP.html rr)rrrr-rérrg@rr.é rgš™™™™™ñ?édég¸…ëQ¸Î?rz Input too bigr-r+ÚstepÚofé2)rÚglobalsr)Úmathr=ÚlogrÚmaxÚ ValueErrorr rrrrr r4rNr Úprintr rÚabsrr) r5ÚverboseZpbitsr7r:ÚsÚMrErFÚqrGÚdr'r'r(Ú npartitionsŠs@   ÿþþ ý rb)F)Z mpmath.libmprrrrrrrr r r r r rrrrrÚsympy.core.numbersrÚresidue_ntheoryrrrrrWr)r4rNrbÚ__all__r'r'r'r(ÚsL V 6