A280444 Least positive integer m such that n - p(m) = x*(3x-1)/2 + y*(3y+1)/2 for some nonnegative integers x and y, or 0 if no such m exists, where p(.) is the partition function given by A000041.

1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 4, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 3, 6, 1, 2, 1, 1, 2, 1, 1, 1, 2, 3, 1, 2, 3, 1, 2, 1, 1, 1, 1, 2, 3, 4, 1, 1, 2, 3, 1, 1, 2, 3, 4, 1, 2, 3, 1, 1, 2, 1, 2, 3, 1, 2, 1, 1, 1

Offset

1,5

Comments

The conjecture in A280455 asserts that a(n) > 0 for all n > 0.

Links

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

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28--Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Example

a(12) = 4 since 12 - p(4) = 12 - 5 = 7 = 0*(3*0-1)/2 + 2*(3*2+1)/2.
a(35) = 6 since 35 - p(6) = 35 - 11 = 24 = 4*(3*4-1)/2 + 1*(3*1+1)/2.
a(4327) = 15 since 4327 - p(15) = 4327 - 176 = 4151 = 16*(3*16-1)/2 + 50*(3*50+1)/2.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
p[n_]:=p[n]=PartitionsP[n];
Pen[n_]:=Pen[n]=SQ[24n+1]&&Mod[Sqrt[24n+1], 6]==1;
Do[m=1; Label[bb]; If[p[m]>n, Goto[cc]]; Do[If[Pen[n-p[m]-x(3x-1)/2], Print[n, " ", m]; Goto[aa]], {x, 0, (Sqrt[24(n-p[m])+1]+1)/6}]; m=m+1; Goto[bb]; Label[cc]; Print[n, " ", 0]; Label[aa]; Continue, {n, 1, 80}]

Crossrefs

Cf. A000041, A000326, A005449, A280455.

Sequence in context: A363279 A205781 A357982 * A030422 A090001 A269329

Adjacent sequences: A280441 A280442 A280443 * A280445 A280446 A280447

Keyword

nonn

Author

Zhi-Wei Sun, Jan 03 2017

Status

approved