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a(0) = 1, and for n > 1, a(n) = (1 + A000120(n))*a(floor(n/2)); also a(n) = A000005(A283477(n)).
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%I #102 Jul 19 2024 22:33:19

%S 1,2,4,6,8,12,18,24,16,24,36,48,54,72,96,120,32,48,72,96,108,144,192,

%T 240,162,216,288,360,384,480,600,720,64,96,144,192,216,288,384,480,

%U 324,432,576,720,768,960,1200,1440,486,648,864,1080,1152,1440,1800,2160,1536,1920,2400,2880,3000

%N a(0) = 1, and for n > 1, a(n) = (1 + A000120(n))*a(floor(n/2)); also a(n) = A000005(A283477(n)).

%H Antti Karttunen, <a href="/A284005/b284005.txt">Table of n, a(n) for n = 0..8191</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%F a(n) = A000005(A283477(n)).

%F Conjecture: a(n) = 2*a(f(n)) + Sum_{k=0..floor(log_2(n))-1} a(f(n) + 2^k*(1 - T(n,k))) for n > 1 with a(0) = 1, a(1) = 2, f(n) = A053645(n), T(n,k) = floor(n/2^k) mod 2. - _Mikhail Kurkov_, Nov 10 2019

%F From _Mikhail Kurkov_, Aug 23 2021: (Start)

%F a(2n+1) = a(n) + a(2n) for n >= 0.

%F a(2n) = a(n) + a(2n - 2^A007814(n)) for n > 0 with a(0) = 1. (End)

%F Conjecture: a(n) = Sum_{k=0..n} (binomial(n, k) mod 2)*A329369(k). In other words, this sequence is modulo 2 binomial transform of A329369. - _Mikhail Kurkov_, Mar 10 2023

%F Conjecture: a(2^m*(2n+1)) = Sum_{k=0..m+1} binomial(m+1, k)*a(2^k*n) for m >= 0, n >= 0 with a(0) = 1. - _Mikhail Kurkov_, Apr 24 2023

%t Table[DivisorSigma[0, #] &@ Apply[Times, Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e == 1 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]] &[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2]], {n, 0, 71}] (* _Michael De Vlieger_, Mar 18 2017 *)

%o (Scheme) (define (A284005 n) (A000005 (A283477 n)))

%o (PARI) A284005(n) = numdiv(A283477(n)); \\ edited by _Michel Marcus_, May 01 2019, _M. F. Hasler_, Nov 10 2019

%o (PARI) a(n) = my(k=if(n,logint(n,2)),s=1); prod(i=0,k, s+=bittest(n,k-i)); \\ _Kevin Ryde_, Jan 20 2021

%Y Cf. A000005, A283477, A284001.

%Y Similar recurrences: A124758, A243499, A329369, A341392.

%K nonn,look

%O 0,2

%A _Antti Karttunen_, Mar 18 2017

%E Made _Mikhail Kurkov_'s Nov 10 2019 formula the new primary name of this sequence - _Antti Karttunen_, Dec 30 2020