A397552 - OEIS

A397552

a(n) = n! mod A000041(n), where A000041 is the partition function.

0, 0, 0, 0, 4, 1, 5, 0, 16, 0, 0, 0, 0, 29, 0, 0, 0, 0, 0, 0, 0, 0, 264, 1185, 0, 110, 420, 2730, 0, 1760, 2328, 308, 0, 0, 9870, 5808, 16416, 2233, 24200, 0, 7938, 36498, 198, 38313, 41075, 85362, 26274, 61446, 71274, 8250, 125818, 218889, 21252, 270963, 52360

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OFFSET

0,5

COMMENTS

The symmetric group has p(n) conjugacy classes and order n!.

The next zero term after a(39), if it exists, is at index n > 2000000. See A046668.

REFERENCES

A. NiShe, Commutativity and Generalizations in Finite Groups, PhD Thesis, University College Cork (2000).

LINKS

Table of n, a(n) for n=0..54.

Robert Heffernan and Des Machale, The genesis of a conjecture in number theory, Irish Math. Soc. Bulletin Number 93, Summer 2024, 39-41 ISSN 0791-5578.

EXAMPLE

a(4) = 4! mod p(4) = 24 mod 5 = 4.

MAPLE

a:= n-> n! mod combinat:-numbpart(n):

seq(a(n), n=0..100);

MATHEMATICA

a[n_] := Mod[n!, PartitionsP[n]]; Array[a, 100]

PROG

(Python) from sympy import factorial, npartitions

def a(n):

return factorial(n) % npartitions(n)

print([a(n) for n in range(1, 101)])