A208251 -id:A208251 - cp's OEIS frontend

A323326 a(n) = 2*T(n) - pi(n), where T(n) (A208251) is the number of refactorable/tau numbers (A033950) <= n and pi(n) (A000720) is the number of primes <= n.

2, 3, 2, 2, 1, 1, 0, 2, 4, 4, 3, 5, 4, 4, 4, 4, 3, 5, 4, 4, 4, 4, 3, 5, 5, 5, 5, 5, 4, 4, 3, 3, 3, 3, 3, 5, 4, 4, 4, 6, 5, 5, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 2, 2, 2, 4, 4, 4, 3, 5, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 2, 4, 3, 3, 3, 3, 3, 3, 2, 4, 4, 4, 3, 5, 5, 5, 5, 7, 6, 6, 6, 6, 6, 6, 6, 8, 7, 7, 7, 7, 6, 6, 5, 7, 7, 7, 6, 8, 7, 7, 7, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 1

Jud McCranie, Jan 11 2019

nonn

Colton conjectured that T(n) >= pi(n)/2 for all n, i.e., this sequence is nonnegative. Zelinsky proved it for n > 7.42*10^13 (see the Zelinsky reference). This calculation went to 7.44*10^13, proving the conjecture.

			For n=6, pi(6)=3, T(6)=2, so a(6) = 2*2 - 3 = 1.

Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999.
Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results , Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.

Cf. A033950, A208251, A000720.

A349322 a(n) = Sum_{d|n} d^c(d), where c is the characteristic function of refactorable numbers (A336040).

Original entry on oeis.org

1, 3, 2, 4, 2, 5, 2, 12, 11, 5, 2, 18, 2, 5, 4, 13, 2, 32, 2, 7, 4, 5, 2, 50, 3, 5, 12, 7, 2, 9, 2, 14, 4, 5, 4, 81, 2, 5, 4, 55, 2, 9, 2, 7, 14, 5, 2, 52, 3, 7, 4, 7, 2, 34, 4, 71, 4, 5, 2, 83, 2, 5, 14, 15, 4, 9, 2, 7, 4, 9, 2, 185, 2, 5, 6, 7, 4, 9, 2, 136, 13, 5, 2, 107

Offset: 1

Wesley Ivan Hurt, Nov 14 2021

nonn

For each divisor d of n, add d if d is refactorable (i.e., if the number of divisors of d divides d), otherwise add 1. For example, the divisors of 8 are 1,2,4,8 and the refactorable divisors of 8 are 1,2,8. The sum is then a(8) = 1 + 2 + 1 + 8 = 12.

Inverse Möbius transform of n^c(n), where c = A336040. - Wesley Ivan Hurt, Jun 29 2024

Antti Karttunen, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Refactorable Number

Cf. A033950, A208251, A335182, A335665, A336040, A336041, A349658.

a[n_] := DivisorSum[n, If[Divisible[#, DivisorSigma[0, #]], #, 1] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)

isrf(n) = n%numdiv(n)==0; \\ A336040
a(n) = sumdiv(n, d, if (isrf(d), d, 1)); \\ Michel Marcus, Nov 16 2021

a(n) = A335182(n) + A349658(n). - Antti Karttunen, Nov 24 2021

a(p) = 2 for odd primes p. - Wesley Ivan Hurt, Nov 28 2021

cp's OEIS Frontend

A323326 a(n) = 2*T(n) - pi(n), where T(n) (A208251) is the number of refactorable/tau numbers (A033950) <= n and pi(n) (A000720) is the number of primes <= n.

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A349322 a(n) = Sum_{d|n} d^c(d), where c is the characteristic function of refactorable numbers (A336040).

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