A329372 Dirichlet convolution of the identity function with A156552.

0, 1, 2, 5, 4, 12, 8, 17, 12, 22, 16, 44, 32, 40, 32, 49, 64, 61, 128, 78, 56, 76, 256, 132, 32, 142, 50, 136, 512, 152, 1024, 129, 104, 274, 88, 209, 2048, 532, 188, 230, 4096, 256, 8192, 252, 148, 1048, 16384, 356, 80, 159, 356, 454, 32768, 240, 160, 392, 680, 2078, 65536, 504, 131072, 4128, 248, 321, 280, 464, 262144, 858, 1328, 400

Offset: 1

Author

Antti Karttunen, Nov 12 2019

Keywords

• nonn

Comments

Equally, Dirichlet convolution of sigma (A000203) with A297112 (Möbius transform of A156552).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..1024
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..16384
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000203, A061395, A156552, A297112, A297167, A329374.

Cf. also A329371, A329373.

Programs

PARI

A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A329372(n) = sumdiv(n,d,(n/d)*A156552(d));

PARI

A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A297112(n) = if(1==n,0,2^A297167(n));
A329372(n) = sumdiv(n,d,sigma(n/d)*A297112(d));

Formula

a(n) = Sum_{d|n} d * A156552(n/d).

a(n) = Sum_{d|n} A000203(n/d) * A297112(d).

A000265(a(n)) = A329374(n).