A245662 -id:A245662 - cp's OEIS frontend

A245525 Unique integer r with -prime(n)/2 < r <= prime(n)/2 such that p(n) == r (mod prime(n)), where p(.) is the partition function given by A000041.

1, -1, -2, -2, -4, -2, -2, 3, 7, 13, -6, 3, 19, 6, -12, 19, 2, 19, 21, -12, -11, -25, 10, -27, 18, 12, 23, -27, -13, -46, -16, -35, 5, -61, -17, 8, -29, -65, -44, -30, 12, -40, 40, -95, 90, 88, 53, 93, 97, -42, -47, 47, 2, 117, -16, 34, 27, 51, -11, 108, -24, 115, -29, 30, -32, -90, -87, 141, 24, 131, -166, -115, -96, -111, 84, -191, 163, -156, 115, 78

Offset: 1

Zhi-Wei Sun, Jul 25 2014

sign

Conjecture: a(n) is always nonzero, i.e., prime(n) never divides the partition number p(n).

This conjecture does not hold with the smallest counterexample being n=1119414 (cf. A245662). - Max Alekseyev, Jul 27 2014

			a(20) = -12 since p(20) = 627 == -12 (mod prime(20)=71).

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A000041, A094252, A245526.

rMod[m_,n_]:=Mod[m,n,-(n-1)/2]
a[n_]:=rMod[PartitionsP[n],Prime[n]]
Table[a[n],{n,1,80}]

a(n) = A094252(n) or A094252(n)-A000040(n), depending on whether A094252(n) <= A000040(n)/2.

A094252 a(n) = partition(n) mod prime(n).

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 15, 3, 7, 13, 25, 3, 19, 6, 35, 19, 2, 19, 21, 59, 62, 54, 10, 62, 18, 12, 23, 80, 96, 67, 111, 96, 5, 78, 132, 8, 128, 98, 123, 143, 12, 141, 40, 98, 90, 88, 53, 93, 97, 187, 186, 47, 2, 117, 241, 34, 27, 51, 266, 108, 259, 115, 278, 30, 281, 227, 244, 141

Offset: 1

Gary W. Adamson, Apr 25 2004

nonn

Indices n such that a(n)=0 (i.e., prime(n) divides partition(n)) are listed in A245662. - Max Alekseyev, Jul 27 2014

			a(10) = 13: partition(10) = 42, prime(10) = 29. 42 mod 29 = 13.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000041, A093952, A245525.

Table[ Mod[ PartitionsP[n], Prime[n]], {n, 70}] (* Robert G. Wilson v, Apr 28 2004 *)

Edited by Robert G. Wilson v, Apr 28 2004

cp's OEIS Frontend

A245525 Unique integer r with -prime(n)/2 < r <= prime(n)/2 such that p(n) == r (mod prime(n)), where p(.) is the partition function given by A000041.

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