A348971 - OEIS

%I #28 Oct 05 2023 04:08:16

%S 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,

%T 22,1,146,13,18,11,60,1,20,15,74,1,30,1,46,36,24,1,146,14,40,19,54,1,

%U 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

%N 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.

%C Möbius transform of A348507.

%H Antti Karttunen, <a href="/A348971/b348971.txt">Table of n, a(n) for n = 1..16384</a>

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

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

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

%t 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 *)

%o (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); };

%o (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); }

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

%Y Cf. A005596, A104141.

%K nonn,easy,look

%O 1,4

%A _Antti Karttunen_, Nov 05 2021