A347136 a(n) = Sum_{d|n} d * A003961(n/d), where A003961 shifts the prime factorization of its argument one step towards larger primes.

1, 5, 8, 19, 12, 40, 18, 65, 49, 60, 24, 152, 30, 90, 96, 211, 36, 245, 42, 228, 144, 120, 52, 520, 109, 150, 272, 342, 60, 480, 68, 665, 192, 180, 216, 931, 78, 210, 240, 780, 84, 720, 90, 456, 588, 260, 100, 1688, 247, 545, 288, 570, 112, 1360, 288, 1170, 336, 300, 120, 1824, 128, 340, 882, 2059, 360, 960, 138

Offset: 1

Author

Antti Karttunen, Aug 24 2021

Keywords

• nonn
• mult

Comments

Dirichlet convolution of the identity function (A000027) with the prime shifted identity (A003961). Multiplicative because both A000027 and A003961 are.

Dirichlet convolution of Euler phi (A000010) with the prime shifted sigma (A003973).

Dirichlet convolution of sigma (A000203) with the prime shifted phi (A003972).

Inverse Möbius transform of A347137.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A003961, A003972, A003973, A151800, A347121, A347137 (Möbius transform).

Cf. also A038040, A336845, A336475, A347236.

Programs

Mathematica

f[p_, e_] := ((np = NextPrime[p])^(e + 1) - p^(e + 1))/(np - p); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 24 2021 *)

PARI

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A347136(n) = sumdiv(n,d,d*A003961(n/d));

Formula

a(n) = Sum_{d|n} d * A003961(n/d).

a(n) = Sum_{d|n} A000010(n/d) * A003973(d).

a(n) = Sum_{d|n} A000203(n/d) * A003972(d).

a(n) = Sum_{d|n} A347137(d).

For all primes p, a(p) = p + A003961(p).

a(n) = A347121(n) + 2*n.

Multiplicative with a(p^e) = (A151800(p)^(e+1) - p^(e+1))/(A151800(p)-p). - Amiram Eldar, Aug 24 2021