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

1,2

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.

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.

Example

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.

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}]

Crossrefs

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

Sequence in context: A131839 A143299 A239428 * A266225 A248596 A076984

Adjacent sequences: A257494 A257495 A257496 * A257498 A257499 A257500

Keyword

nonn

Author

Zhi-Wei Sun, Apr 26 2015

Status

approved