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A280455 Number of ways to write n as x*(3x-1)/2 + y*(3y+1)/2 + p(z), where x,y,z are nonnegative integers with z > 0, and p(.) is the partition function given by A000041. 4
1, 2, 3, 3, 3, 3, 3, 5, 4, 5, 2, 4, 4, 5, 5, 4, 6, 5, 5, 3, 4, 6, 7, 3, 5, 3, 8, 3, 8, 7, 6, 5, 4, 7, 3, 4, 6, 8, 4, 5, 4, 12, 5, 8, 5, 6, 4, 5, 8, 5, 4, 7, 7, 6, 5, 7, 8, 5, 9, 6, 6, 5, 10, 8, 6, 3, 7, 8, 7, 4, 6, 7, 9, 3, 5, 4, 8, 7, 9, 13 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Conjecture: (i) a(n) > 0 for all n > 0.

(ii) lim_n a(n)/(log n)^2 = 1/Pi^2.

This is similar to the author's conjecture in A280386. At the author's request, Prof. Qing-Hu Hou at Tianjin Univ. has verified part (i) of the above conjecture for n up to 10^9.

We also have some other similar conjectures. For example, we conjecture that any positive integer can be expressed as the sum of two triangular numbers and a partition number.

As the main term of log p(n) is Pi*sqrt(2n/3), the partition function p(n) eventually grows faster than any polynomial.

See also A280472 for a similar conjecture.

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(1) = 1 since 1 = 0*(3*0-1)/2 + 0*(3*0+1)/2 + p(1).

a(2) = 2 since 2 = 1*(3*1-1)/2 + 0*(3*0+1)/2 + p(1) = 0*(3*0-1)/2 + 0*(3*0+1)/2 + p(2).

a(2771) = 1 since 2771 = 35*(3*35-1)/2 + 25*(3*25+1)/2 + p(1).

a(9426) = 1 since 9426 = 4*(3*4-1)/2 + 79*(3*79+1)/2 + p(3).

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

CROSSREFS

Cf. A000041, A000217, A000326, A005449, A280386, A280444, A280472.

Sequence in context: A246262 A275974 A052288 * A284725 A055767 A029110

Adjacent sequences:  A280452 A280453 A280454 * A280456 A280457 A280458

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jan 03 2017

STATUS

approved

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Last modified September 26 09:15 EDT 2021. Contains 347664 sequences. (Running on oeis4.)