A236765 Number of ways to write n = k^2 + m with k > 1 and m > 1 such that sigma(k^2) + prime(m) - 1 is prime, where sigma(j) denotes the sum of all positive divisors of j.

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 0, 2, 3, 1, 3, 2, 1, 2, 2, 3, 2, 4, 3, 2, 3, 2, 2, 3, 3, 2, 2, 2, 3, 4, 4, 3, 4, 4, 1, 3, 4, 2, 2, 5, 3, 3, 4, 4, 3, 1, 5, 3, 4, 3, 4, 5, 4, 3, 1, 5, 2, 6, 4, 3, 4, 2, 1, 5, 4, 7, 4, 4, 3, 1, 3, 1, 4, 4, 4, 2, 5, 6, 3, 6, 5, 5, 1, 4, 5, 5, 4, 3, 6

Offset

1,13

Comments

Conjecture: (i) If n > 6 is not equal to 18, then a(n) > 0.

(ii) Any integer n > 14 can be written as p + q with q > 0 such that p, p + 6 and prime(p) + sigma(q) are all prime.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Maple

a(10) = 1 since 10 = 2^2 + 6 with sigma(2^2) + prime(6) - 1 = 7 + 13 - 1 = 19 prime.
a(253) = 1 since 253 = 15^2 + 28 with sigma(15^2) + prime(28) - 1 = 403 + 107 - 1 = 509 prime.

Mathematica

p[n_, k_]:=PrimeQ[DivisorSigma[1, k^2]+Prime[n-k^2]-1]
a[n_]:=If[n<6, 0, Sum[If[p[n, k], 1, 0], {k, 2, Sqrt[n-2]}]]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000040, A000203, A233544, A233654, A233793, A233864, A236548.

Sequence in context: A349277 A307014 A240871 * A171531 A171532 A171533

Adjacent sequences: A236762 A236763 A236764 * A236766 A236767 A236768

Keyword

nonn

Author

Zhi-Wei Sun, Jan 30 2014

Status

approved