A353748 a(n) = phi(n) - A064989(n), where phi is Euler totient function, and A064989 shifts the prime factorization one step towards lower primes.

0, 0, 0, 1, 1, 0, 1, 3, 2, 1, 3, 2, 1, 1, 2, 7, 3, 2, 1, 5, 2, 3, 3, 6, 11, 1, 10, 7, 5, 2, 1, 15, 6, 3, 9, 8, 5, 1, 2, 13, 3, 2, 1, 13, 12, 3, 3, 14, 17, 11, 6, 13, 5, 10, 19, 19, 2, 5, 5, 10, 1, 1, 16, 31, 15, 6, 5, 19, 6, 9, 3, 20, 1, 5, 22, 19, 25, 2, 5, 29, 38, 3, 3, 14, 25, 1, 10, 33, 5, 12, 17, 25, 2, 3, 21

Offset

1,8

Links

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

Index entries for sequences computed from indices in prime factorization.

Formula

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

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

For n > 1, a(n) = A140434(n) - A353747(n).

Sum_{k=1..n} a(k) ~ c * n^2, where c = 3/Pi^2 - (1/2) * Product_{p prime} ((p^2-p)/(p^2-q(p))) = 0.0832596219... , where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Dec 21 2023

Mathematica

f1[p_, e_] := (p - 1)*p^(e - 1); f2[p_, e_] := If[p == 2, 1, NextPrime[p, -1]^e]; a[1] = 0; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) - Times @@ f2 @@@ f; Array[a, 100] (* 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); };
A353748(n) = (eulerphi(n)-A064989(n));

Crossrefs

Cf. A000010, A064989, A140434, A151799, A353747.

Sequence in context: A130827 A070309 A287556 * A236966 A280048 A119910

Adjacent sequences: A353745 A353746 A353747 * A353749 A353750 A353751

Keyword

nonn

Author

Antti Karttunen, May 06 2022

Status

approved