A256071 Number of ordered ways to write n = p + x*(3*x-1)/2, where p is prime or zero, and x is an integer.

1, 1, 2, 2, 2, 3, 1, 4, 2, 2, 2, 1, 4, 2, 3, 3, 1, 3, 4, 3, 3, 1, 3, 2, 4, 3, 3, 1, 3, 4, 2, 4, 2, 3, 2, 3, 2, 3, 4, 3, 2, 3, 4, 5, 3, 4, 3, 2, 4, 3, 1, 3, 3, 5, 4, 3, 2, 3, 4, 5, 3, 2, 4, 4, 4, 2, 3, 2, 5, 4, 3, 3, 4, 5, 5, 3, 4, 3, 3, 4, 5, 4, 4, 5, 3, 3, 3, 3, 6, 3, 3, 2, 2, 4, 7, 3, 3, 3, 4, 5, 3

Offset: 0

Author

Zhi-Wei Sun, Mar 13 2015

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n. Moreover, each nonnegative integer n is either an odd prime, or a generalized pentagonal number, or the sum of an odd prime and a generalized pentagonal number.

This is similar to the author's earlier conjecture on sums of primes and triangular numbers (see the reference and also A132399).

The conjecture has been verified for all n = 0..10^9.

Examples

			a(11) = 1 since 11 = 11 + 0*(3*0-1)/2 with 11 prime.
a(15) = 1 since 15 = 0 + (-3)*(3*(-3)-1)/2.
a(50) = 1 since 50 = 43 + (-2)*(3*(-2)-1)/2 with 43 prime.
		

References

• Zhi-Wei Sun, On sums of primes and triangular numbers, J. Comb. Number Theory 1(2009), 65-76.

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
• Zhi-Wei Sun, On sums of primes and triangular numbers, arXiv:0803.3737 [math.NT], 2008-2009.
• Zhi-Wei Sun, On universal sums of polygonal numbers, arXiv:0905.0635 [math.NT], 2009-2015.

Crossrefs

Cf. A000040, A001318, A132399, A256119.

Programs

Mathematica

P[n_]:=(n==0)||PrimeQ[n]
Do[r=0;Do[If[P[n-x(3x-1)/2],r=r+1],{x,-Floor[(Sqrt[24n+1]-1)/6],Floor[(Sqrt[24n+1]+1)/6]}];Print[n," ",r];Label[aa];Continue,{n,0,100}]

PARI

a(n)=if(n==0, return(1)); sum(x=1,(1+sqrt(24*n+1))\6, isprime(n-x*(3*x-1)/2))+sum(x=0,(sqrt(24*n+1)-1)\6, isprime(n-x*(3*x+1)/2))+ispolygonal(n,5)+(x->3*x^2+x==2*n)(round((sqrt(24*n+1)-1)/6)) \\ Charles R Greathouse IV, Apr 07 2015