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

1,2

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

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

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 *)

Prog

(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));

Crossrefs

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

Cf. also A038040, A336845, A336475, A347236.

Sequence in context: A302393 A342804 A226902 * A326826 A026595 A037233

Adjacent sequences: A347133 A347134 A347135 * A347137 A347138 A347139

Keyword

nonn,mult

Author

Antti Karttunen, Aug 24 2021

Status

approved