A353747 a(n) = phi(n) + A064989(n), where phi is Euler totient function, and A064989 shifts the prime factorization one step towards lower primes.

2, 2, 4, 3, 7, 4, 11, 5, 10, 7, 17, 6, 23, 11, 14, 9, 29, 10, 35, 11, 22, 17, 41, 10, 29, 23, 26, 17, 51, 14, 59, 17, 34, 29, 39, 16, 67, 35, 46, 19, 77, 22, 83, 27, 36, 41, 89, 18, 67, 29, 58, 35, 99, 26, 61, 29, 70, 51, 111, 22, 119, 59, 56, 33, 81, 34, 127, 45, 82, 39, 137, 28, 143, 67, 58, 53, 95, 46, 151, 35, 70

Offset: 1

Author

Antti Karttunen, May 06 2022

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences computed from indices in prime factorization.

Crossrefs

Cf. A000010, A064989, A104141, A140434, A151799, A353748.

Programs

Mathematica

f1[p_, e_] := (p - 1)*p^(e - 1); f2[p_, e_] := If[p == 2, 1, NextPrime[p, -1]^e]; a[1] = 2; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) + Times @@ f2 @@@ f; Array[a, 80] (* Amiram Eldar, May 07 2022 *)

PARI

A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A353747(n) = (eulerphi(n)+A064989(n));

Formula

a(n) = A000010(n) + A064989(n).

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

For n > 1, a(n) = A140434(n) - A353748(n).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2-p)/(p^2-q(p))) + 3/Pi^2 = 0.524667479..., where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Dec 21 2023