A285705 a(n) = 2*n - A285703(n) = 2*n - A000203(A064216(n)).

1, 1, 2, 2, 3, 4, 2, 4, 4, 2, 4, 4, 13, 13, 6, 2, 10, 12, 6, 4, 4, 2, 18, 4, 19, 10, 6, 24, 4, 6, 2, 22, 18, 6, 10, 4, 2, 37, 30, 6, 51, 4, 30, 16, 6, 20, 4, 24, 8, 44, 4, 2, 34, 4, 2, 16, 4, 36, 34, 36, 65, 10, 86, 14, 4, 4, 26, 76, 6, 2, 10, 48, 50, 55, 10, 2, 56, 36, 6, 16, 42, 6, 70, 4, 37, 46, 6, 98, 16, 6, 2, 4, 58, 76, 100, 10, 2, 52, 4, 2, 16, 60, 54

Offset: 1

Author

Antti Karttunen, Apr 26 2017

Keywords

• nonn
• hear

Comments

Question: Are all terms positive? - Yes, they are, see A286385. (Note added Jul 24 2022).

For listening: fast tempo and percussive instrument, default "modulo 88" pitch mapping, all 10000 terms.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A033879, A064989, A064216, A151799, A285703, A285704, A286385.

Programs

Mathematica

Table[2 n - DivisorSigma[1, #] &@ If[n == 1, 1, Apply[Times, FactorInteger[2 n - 1] /. {p_, e_} /; p > 2 :> NextPrime[p, -1]^e]], {n, 103}] (* Michael De Vlieger, Apr 26 2017 *)

PARI

A064989(n) = { my(f = factor(n)); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A285705(n) = (n+n - sigma(A064989(n+n-1))); \\ Antti Karttunen, Jul 24 2022

Scheme

(define (A285705 n) (- (* 2 n) (A285703 n)))

Formula

a(n) = 2*n - A285703(n) = 2*n - A000203(A064216(n)).

a(n) = 1 + A286385(A064216(n)). - Antti Karttunen, Jul 24 2022

Sum_{k=1..n} a(k) ~ c * n^2, where c = 1 - Product_{p prime} (p^3/((p+1)*(p^2-q(p)))) = 0.1831523243..., where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Dec 21 2023