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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.
5
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
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); }
KEYWORD
nonn,easy,look
AUTHOR
Antti Karttunen, Nov 05 2021
STATUS
approved