U Zʹj#@sddlmZmZddlmZddlmZddlmZe e dddZ e dd d Z e dd d Z e dd dZddZddZdS))chain combinations)gcd) factorint)as_int)factorsreturncCsT|D]F}|D]8\}}d}t|D]"}|||}|dkr(dSq(qqdS)z Check whether `n` is a nilpotent number. Note that ``factors`` is a prime factorization of `n`. This is a low-level helper for ``is_nilpotent_number``, for internal use. FT)keysitemsrange)rpqem_r`/var/www/html/bwave/python/venv/lib/python3.8/site-packages/sympy/combinatorics/group_numbers.py_is_nilpotent_numbers   r)rcCs(t|}|dkrtd|tt|S)aj Check whether `n` is a nilpotent number. A number `n` is said to be nilpotent if and only if every finite group of order `n` is nilpotent. For more information see [1]_. Examples ======== >>> from sympy.combinatorics.group_numbers import is_nilpotent_number >>> from sympy import randprime >>> is_nilpotent_number(21) False >>> is_nilpotent_number(randprime(1, 30)**12) True References ========== .. [1] Pakianathan, J., Shankar, K., Nilpotent Numbers, The American Mathematical Monthly, 107(7), 631-634. .. [2] https://oeis.org/A056867 r$n must be a positive integer, not %i)r ValueErrorrr)nrrris_nilpotent_numbers rcCsBt|}|dkrtd|t|}tdd|Do@t|S)a Check whether `n` is an abelian number. A number `n` is said to be abelian if and only if every finite group of order `n` is abelian. For more information see [1]_. Examples ======== >>> from sympy.combinatorics.group_numbers import is_abelian_number >>> from sympy import randprime >>> is_abelian_number(4) True >>> is_abelian_number(randprime(1, 2000)**2) True >>> is_abelian_number(60) False References ========== .. [1] Pakianathan, J., Shankar, K., Nilpotent Numbers, The American Mathematical Monthly, 107(7), 631-634. .. [2] https://oeis.org/A051532 rrcss|]}|dkVqdS)Nr.0rrrr Vsz$is_abelian_number..rrrallvaluesrrrrrris_abelian_number8s  r!cCsBt|}|dkrtd|t|}tdd|Do@t|S)a Check whether `n` is a cyclic number. A number `n` is said to be cyclic if and only if every finite group of order `n` is cyclic. For more information see [1]_. Examples ======== >>> from sympy.combinatorics.group_numbers import is_cyclic_number >>> from sympy import randprime >>> is_cyclic_number(15) True >>> is_cyclic_number(randprime(1, 2000)**2) False >>> is_cyclic_number(4) False References ========== .. [1] Pakianathan, J., Shankar, K., Nilpotent Numbers, The American Mathematical Monthly, 107(7), 631-634. .. [2] https://oeis.org/A003277 rrcss|]}|dkVqdSr Nrrrrrrwsz#is_cyclic_number..rr rrris_cyclic_numberYs  r#csfddD}|d}tfddttdD}|D]`}t|}d}|D]>tfdd||BD}||dd9}|s^qq^||7}qF|S) a| Number of groups of order `n`. where `n` is squarefree and its prime factors are ``prime_factors``. i.e., ``n == math.prod(prime_factors)`` Explanation =========== When `n` is squarefree, the number of groups of order `n` is expressed by .. math :: \sum_{d \mid n} \prod_p \frac{p^{c(p, d)} - 1}{p - 1} where `n=de`, `p` is the prime factor of `e`, and `c(p, d)` is the number of prime factors `q` of `d` such that `q \equiv 1 \pmod{p}` [2]_. The formula is elegant, but can be improved when implemented as an algorithm. Since `n` is assumed to be squarefree, the divisor `d` of `n` can be identified with the power set of prime factors. We let `N` be the set of prime factors of `n`. `F = \{p \in N : \forall q \in N, q \not\equiv 1 \pmod{p} \}, M = N \setminus F`, we have the following. .. math :: \sum_{d \in 2^{M}} \prod_{p \in M \setminus d} \frac{p^{c(p, F \cup d)} - 1}{p - 1} Practically, many prime factors are expected to be members of `F`, thus reducing computation time. Parameters ========== prime_factors : set The set of prime factors of ``n``. where `n` is squarefree. Returns ======= int : Number of groups of order ``n`` Examples ======== >>> from sympy.combinatorics.group_numbers import _holder_formula >>> _holder_formula({2}) # n = 2 1 >>> _holder_formula({2, 3}) # n = 2*3 = 6 2 See Also ======== groups_count References ========== .. [1] Otto Holder, Die Gruppen der Ordnungen p^3, pq^2, pqr, p^4, Math. Ann. 43 pp. 301-412 (1893). http://dx.doi.org/10.1007/BF01443651 .. [2] John H. Conway, Heiko Dietrich and E.A. O'Brien, Counting groups: gnus, moas and other exotica The Mathematical Intelligencer 30, 6-15 (2008) https://doi.org/10.1007/BF02985731 cs&h|]tfddDrqS)c3s|]}|dkVqdSr"rrrr rrrsz,_holder_formula...)r)r) prime_factorsr%r sz"_holder_formula..rc3s|]}t|VqdS)N)r)rr)Mrrrsz"_holder_formula..r csg|]}|dkr|qS)r rr$r%rr s z#_holder_formula..)r from_iterabler lenset)r&FsZpowersetZpsprodcr)r)r r&r_holder_formulazs?$  r2cCs t|}|dkrtd|t|}t|dkrJt|d\}}|dkr|dddddddd d d d g }|t|kr|||S|d krddddddddddg }|t|kr||S|dkr|S|d krdS|dkrdS|dkr dd|dt|dd t|ddS|dkrbd |dd|ddt|dd dt|dddt|ddS|dkrJ|dkrzdSd |dd|dd|d d|dd |d!d|dd|d"t|dd |dd#|d$t|ddd |d%t|dddt|dddt|dd&t|dd'Std(d)|Drdddddddddd*dddd+d+ddd,d dd*ddd-dd.}||kr||Std/t|dkrt| \}}||kr||}}||dkrdSdSt t | S)0a Number of groups of order `n`. In [1]_, ``gnu(n)`` is given, so we follow this notation here as well. Parameters ========== n : Integer ``n`` is a positive integer Returns ======= int : ``gnu(n)`` Raises ====== ValueError Number of groups of order ``n`` is unknown or not implemented. For example, gnu(`2^{11}`) is not yet known. On the other hand, gnu(12) is known to be 5, but this has not yet been implemented in this function. Examples ======== >>> from sympy.combinatorics.group_numbers import groups_count >>> groups_count(3) # There is only one cyclic group of order 3 1 >>> # There are two groups of order 10: the cyclic group and the dihedral group >>> groups_count(10) 2 See Also ======== is_cyclic_number `n` is cyclic iff gnu(n) = 1 References ========== .. [1] John H. Conway, Heiko Dietrich and E.A. O'Brien, Counting groups: gnus, moas and other exotica The Mathematical Intelligencer 30, 6-15 (2008) https://doi.org/10.1007/BF02985731 .. [2] https://oeis.org/A000001 rrr 3i i ii!lyLZ .rCii^$imMlNC='iX i ,ii i# css|]}|dkVqdSr"rrrrrrszgroups_count..4 2 )r@r=$(rA-0rJrH68<?DHKLPTXZ\z9Number of groups of order n is unknown or not implemented) rrrr,listr ranyrr r2r-)rrr rZA000679ZA090091Zsmallrrrr groups_counts2    ,   :$     rbN) itertoolsrrZsympy.external.gmpyrZsympy.ntheory.factor_rZsympy.utilities.miscrdictboolrrr!r#r2rbrrrrs   !!P