A257497 Number of ordered ways to write n as the sum of a term of A257121 and a positive generalized pentagonal number.

1, 2, 2, 2, 3, 2, 3, 3, 3, 2, 2, 2, 5, 2, 3, 4, 4, 4, 2, 2, 3, 4, 6, 3, 2, 5, 7, 5, 2, 4, 3, 5, 4, 3, 4, 4, 6, 5, 3, 3, 5, 4, 5, 2, 2, 5, 4, 4, 2, 3, 5, 5, 6, 1, 4, 5, 4, 3, 3, 7, 4, 2, 5, 2, 5, 4, 2, 4, 3, 6, 4, 5, 9, 4, 3, 3, 4, 8, 2, 4, 5, 3, 5, 1, 5, 4, 1, 5, 3, 2, 4, 6, 6, 3, 5, 4, 6, 5, 5, 5

Offset: 1

Author

Zhi-Wei Sun, Apr 26 2015

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 54, 84, 87, 109, 174, 252, 344, 1234, 1439, 2924.

This implies the Twin Prime Conjecture.

Examples

			a(1439) = 1 since 1439 = 1424 + 15 = floor(4274/3) + (-3)*(3*(-3)-1)/2 with {3*4274-1,3*4274+1} = {12821,12823} a twin prime pair.
a(2924) = 1 since 2924 = 2334 + 590 = floor(7004/3) + 20*(3*20-1)/2 with {3*7004-1, 3*7004+1} = {21011,21013} a twin prime pair.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..20000
• Zhi-Wei Sun, Natural numbers represented by floor(x^2/a) + floor(y^2/b) + floor(z^2/c), arXiv:1504.01608 [math.NT], 2015.

Crossrefs

Cf. A001318, A014574, A257121, A256071, A256707, A257317, A257474.

Programs

Mathematica

TQ[n_]:=PrimeQ[3n-1]&&PrimeQ[3n+1]
PQ[n_]:=TQ[3*n]||TQ[3*n+1]||TQ[3n+2]
SQ[n_]:=IntegerQ[Sqrt[24n+1]]
Do[m=0;Do[If[PQ[x]&&SQ[n-x],m=m+1],{x,0,n-1}];
Print[n," ",m];Continue,{n,1,100}]