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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A266476 A081327 A205781 * A030422 A090001 A269329

Adjacent sequences:  A280441 A280442 A280443 * A280445 A280446 A280447

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jan 03 2017

STATUS

approved

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Last modified March 24 12:14 EDT 2019. Contains 321448 sequences. (Running on oeis4.)