A348971 a(n) = Product(p*(p+1)^(e-1)) - Product((p-1)*p^(e-1)), when n = Product(p^e), with p primes, and e their exponents.

0, 1, 1, 4, 1, 4, 1, 14, 6, 6, 1, 14, 1, 8, 7, 46, 1, 18, 1, 22, 9, 12, 1, 46, 10, 14, 30, 30, 1, 22, 1, 146, 13, 18, 11, 60, 1, 20, 15, 74, 1, 30, 1, 46, 36, 24, 1, 146, 14, 40, 19, 54, 1, 78, 15, 102, 21, 30, 1, 74, 1, 32, 48, 454, 17, 46, 1, 70, 25, 46, 1, 192, 1, 38, 50, 78, 17, 54, 1, 238, 138, 42, 1, 102, 21

Offset

1,4

Comments

Möbius transform of A348507.

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Formula

a(n) = A003968(n) - A000010(n).

a(n) = Sum_{d|n} A008683(n/d) * A348507(d).

Sum_{k=1..n} a(k) ~ c * n^2, where c = A104141 * (1/A005596 - 1) = 0.5088692487... . - Amiram Eldar, Oct 05 2023

Mathematica

f1[p_, e_] := p*(p + 1)^(e - 1); f2[p_, e_] := (p - 1)*p^(e - 1); a[1] = 0; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) - Times @@ f2 @@@ f; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)

Prog

(PARI) A348971(n) = { my(f=factor(n), m1=1, m2=1, p); for(i=1, #f~, p = f[i, 1]; m1 *= p*(p+1)^(f[i, 2]-1); m2 *= (p-1)*p^(f[i, 2]-1)); (m1-m2); };
(PARI) A348971(n) = { my(f=factor(n), p); for (i=1, #f~, p = f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f)-eulerphi(n); }

Crossrefs

Cf. A000010, A003959, A003968, A008683, A300251, A348507, A348970.

Cf. A005596, A104141.

Sequence in context: A301598 A077088 A376132 * A358821 A297420 A156896

Adjacent sequences: A348968 A348969 A348970 * A348972 A348973 A348974

Keyword

nonn,easy,look

Author

Antti Karttunen, Nov 05 2021

Status

approved