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A353749
a(n) = phi(n) * A064989(n), where phi is Euler totient function, and A064989 shifts the prime factorization one step towards lower primes.
15
1, 1, 4, 2, 12, 4, 30, 4, 24, 12, 70, 8, 132, 30, 48, 8, 208, 24, 306, 24, 120, 70, 418, 16, 180, 132, 144, 60, 644, 48, 870, 16, 280, 208, 360, 48, 1116, 306, 528, 48, 1480, 120, 1722, 140, 288, 418, 1978, 32, 1050, 180, 832, 264, 2444, 144, 840, 120, 1224, 644, 3074, 96, 3540, 870, 720, 32, 1584, 280, 4026, 416
OFFSET
1,3
FORMULA
Multiplicative with a(p^e) = (p - 1) * p^(e-1) * q^e, where q is the largest prime less than p, and 1 if p = 2.
a(n) = A000010(n) * A064989(n).
For n >= 0, a(4n+2) = a(2n+1).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/Pi^2) * Product_{p prime} ((p^3-q)/((p+1)*(p^2-q))) = 0.1118576617..., where q(p) = nextprime(p) = A151800(p). - Amiram Eldar, Dec 31 2022
MATHEMATICA
f[p_, e_] := (p - 1)*p^(e - 1)*If[p == 2, 1, NextPrime[p, -1]^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 70] (* Amiram Eldar, May 07 2022 *)
PROG
(PARI)
A064989(n) = { my(f=factor(n>>valuation(n, 2))); for(i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f); };
A353749(n) = (eulerphi(n)*A064989(n));
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Antti Karttunen, May 07 2022
STATUS
approved