A347122 -id:A347122 - cp's OEIS frontend

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

Antti Karttunen, Aug 24 2021

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.

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

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

Cf. also A018804, A347237.

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 *)

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));

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

A347121 a(n) = A347136(n) - 2*n.

Original entry on oeis.org

-1, 1, 2, 11, 2, 28, 4, 49, 31, 40, 2, 128, 4, 62, 66, 179, 2, 209, 4, 188, 102, 76, 6, 472, 59, 98, 218, 286, 2, 420, 6, 601, 126, 112, 146, 859, 4, 134, 162, 700, 2, 636, 4, 368, 498, 168, 6, 1592, 149, 445, 186, 466, 6, 1252, 178, 1058, 222, 184, 2, 1704, 6, 216, 756, 1931, 230, 828, 4, 548, 278, 940, 2, 3041, 6

Offset: 1

Antti Karttunen, Aug 24 2021

sign

Cf. A000040, A001223, A003961, A347136, A347122 (Möbius transform).

Cf. also A341512, A346239, A347236.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A347121(n) = (sumdiv(n,d,d*A003961(n/d))-(2*n));

a(n) = A347136(n) - 2*n.

a(A000040(n)) = A001223(n).

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula