A353747 - OEIS

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

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OFFSET

1,1

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) - 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

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 *)

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); };

A353747(n) = (eulerphi(n)+A064989(n));

CROSSREFS

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

Sequence in context: A374405 A336305 A265701 * A007439 A238622 A096441

Adjacent sequences: A353744 A353745 A353746 * A353748 A353749 A353750

KEYWORD

nonn

AUTHOR

Antti Karttunen, May 06 2022

STATUS

approved