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

1,2

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.

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

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 *)

Prog

(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));

Crossrefs

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

Cf. also A018804, A347237.

Sequence in context: A074862 A101064 A243707 * A347413 A310825 A062380

Adjacent sequences: A347134 A347135 A347136 * A347138 A347139 A347140

Keyword

nonn,easy,mult

Author

Antti Karttunen, Aug 24 2021

Status

approved