A353749 - OEIS

%I #19 Dec 31 2022 03:52:48

%S 1,1,4,2,12,4,30,4,24,12,70,8,132,30,48,8,208,24,306,24,120,70,418,16,

%T 180,132,144,60,644,48,870,16,280,208,360,48,1116,306,528,48,1480,120,

%U 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

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

%H Antti Karttunen, <a href="/A353749/b353749.txt">Table of n, a(n) for n = 1..16384</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.

%F 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.

%F a(n) = A000010(n) * A064989(n).

%F For n >= 0, a(4n+2) = a(2n+1).

%F 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

%t 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 *)

%o (PARI)

%o A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };

%o A353749(n) = (eulerphi(n)*A064989(n));

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

%K nonn,mult

%O 1,3

%A _Antti Karttunen_, May 07 2022