A347121 -id:A347121 - cp's OEIS frontend

A347122 Möbius transform of A347121.

-1, 2, 3, 10, 3, 24, 5, 38, 29, 36, 3, 90, 5, 56, 61, 130, 3, 152, 5, 138, 95, 72, 7, 306, 57, 92, 187, 214, 3, 292, 7, 422, 121, 108, 139, 550, 5, 128, 155, 474, 3, 452, 5, 282, 403, 160, 7, 990, 145, 348, 181, 358, 7, 856, 173, 734, 215, 180, 3, 1046, 7, 208, 625, 1330, 223, 604, 5, 426, 269, 700, 3, 1838, 7, 236

Offset: 1

Antti Karttunen, Aug 24 2021

sign

Cf. A000010, A001223, A003961, A008683, A347121, A347136, A347137.

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));
A347122(n) = sumdiv(n,d,moebius(n/d)*A347121(d));

a(n) = Sum_{d|n} A008683(n/d) * A347121(d).
a(n) = A347137(n) - 2*A000010(n).
For all n >= 1, a(A000040(n)) = 1 + A001223(n).

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

Original entry on oeis.org

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

Antti Karttunen, Aug 24 2021

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.

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

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

Cf. also A038040, A336845, A336475, A347236.

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 *)

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));

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

cp's OEIS Frontend

A347122 Möbius transform of A347121.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula