A353749 a(n) = phi(n) * A064989(n), where phi is Euler totient function, and A064989 shifts the prime factorization one step towards lower primes.

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

Links

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

Index entries for sequences computed from indices in prime factorization.

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

Cf. A000010, A064989, A151800, A353747, A353748, A353750.

Sequence in context: A226725 A354190 A137447 * A077015 A077016 A191436

Adjacent sequences: A353746 A353747 A353748 * A353750 A353751 A353752

Keyword

nonn,mult

Author

Antti Karttunen, May 07 2022

Status

approved