A347137 a(n) = Sum_{d|n} phi(d) * A003961(n/d), where A003961 shifts the prime factorization of its argument one step towards larger primes, and phi is Euler totient function.

1, 4, 7, 14, 11, 28, 17, 46, 41, 44, 23, 98, 29, 68, 77, 146, 35, 164, 41, 154, 119, 92, 51, 322, 97, 116, 223, 238, 59, 308, 67, 454, 161, 140, 187, 574, 77, 164, 203, 506, 83, 476, 89, 322, 451, 204, 99, 1022, 229, 388, 245, 406, 111, 892, 253, 782, 287, 236, 119, 1078, 127, 268, 697, 1394, 319, 644, 137, 490

Offset: 1

Author

Antti Karttunen, Aug 24 2021

Keywords

• nonn
• easy
• mult

Comments

Dirichlet convolution of Euler phi (A000010) with the prime shift function (A003961). Multiplicative because both A000010 and A003961 are.

Dirichlet convolution of the identity function (A000027) with the prime shifted phi (A003972).

Möbius transform of A347136.

Links

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

Crossrefs

Cf. A000010, A000027, A000040, A001043, A003961, A003972, A008683, A151800, A347122, A347136 (inverse Möbius transform).

Cf. also A018804, A347237.

Programs

Mathematica

f[p_, e_] := (q = NextPrime[p])^e + (p - 1)*(q^e - p^e)/(q - p); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2023 *)

PARI

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A347137(n) = sumdiv(n,d,eulerphi(n/d)*A003961(d));

Formula

a(n) = Sum_{d|n} A000010(n/d) * A003961(d).

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

a(n) = Sum_{d|n} A008683(n/d) * A347136(d).

a(n) = A347122(n) + 2*A000010(n).

a(A000040(n)) = A001043(n) - 1.

Multiplicative with a(p^e) = q(p)^e + (p-1)*(q(p)^e - p^e)/(q(p) - p), where q(p) = nextprime(p) = A151800(p). - Amiram Eldar, Sep 16 2023