A345298 a(n) = Sum_{p|n, p prime} tau(p #).

0, 2, 4, 2, 8, 6, 16, 2, 4, 10, 32, 6, 64, 18, 12, 2, 128, 6, 256, 10, 20, 34, 512, 6, 8, 66, 4, 18, 1024, 14, 2048, 2, 36, 130, 24, 6, 4096, 258, 68, 10, 8192, 22, 16384, 34, 12, 514, 32768, 6, 16, 10, 132, 66, 65536, 6, 40, 18, 260, 1026, 131072, 14, 262144, 2050, 20, 2, 72

Offset

1,2

Comments

If p is prime, a(p) = Sum_{p|p} tau(p #) = tau(p) * tau(prevprime(p)) * ... * tau(2) = 2 * 2 * ... * 2 ( pi(p) times ) = 2^pi(p).

Links

Table of n, a(n) for n=1..65.

Formula

G.f.: Sum_{k>=1} 2^k * x^prime(k) / (1 - x^prime(k)). - Ilya Gutkovskiy, Aug 18 2021

Example

a(14) = Sum_{p|14} tau(p #) = tau(2 #) + tau(7 #) = 2^pi(2) + 2^pi(7) = 2^1 + 2^4 = 18.

Mathematica

Table[Sum[DivisorSigma[0, Product[i^(PrimePi[i] - PrimePi[i - 1]), {i, k}]](PrimePi[k] - PrimePi[k - 1]) (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 80}]

Crossrefs

Equals twice A087207.

Cf. A000005 (tau), A002110, A345284.

Sequence in context: A204898 A240295 A113477 * A279350 A278221 A287879

Adjacent sequences: A345295 A345296 A345297 * A345299 A345300 A345301

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 13 2021

Status

approved