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%I #26 Dec 21 2023 10:22:47
%S 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,
%T 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,
%U 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
%N a(n) = 2*n - A285703(n) = 2*n - A000203(A064216(n)).
%C Question: Are all terms positive? - Yes, they are, see A286385. (Note added Jul 24 2022).
%C For listening: fast tempo and percussive instrument, default "modulo 88" pitch mapping, all 10000 terms.
%H Antti Karttunen, <a href="/A285705/b285705.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = 2*n - A285703(n) = 2*n - A000203(A064216(n)).
%F a(n) = 1 + A286385(A064216(n)). - _Antti Karttunen_, Jul 24 2022
%F 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
%t 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 *)
%o (PARI)
%o 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); };
%o A285705(n) = (n+n - sigma(A064989(n+n-1))); \\ _Antti Karttunen_, Jul 24 2022
%o (Scheme) (define (A285705 n) (- (* 2 n) (A285703 n)))
%Y Cf. A000203, A033879, A064989, A064216, A151799, A285703, A285704, A286385.
%K nonn,hear
%O 1,3
%A _Antti Karttunen_, Apr 26 2017
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