cp's OEIS Frontend

A353484 a(1) = 0; and for n > 1, a(n) = A165560(n) + a(A064989(n)), where A165560 is the parity of arithmetic derivative, and A064989 shifts the prime factorization of its argument one step toward lower primes.

%I A353484 #19 Apr 23 2022 09:41:30
%S A353484 0,1,2,0,3,2,4,0,0,3,5,1,6,4,2,0,7,1,8,2,3,5,9,1,0,6,1,3,10,3,11,0,4,
%T A353484 7,2,0,12,8,5,2,13,4,14,4,2,9,15,1,0,1,6,5,16,1,3,3,7,10,17,2,18,11,3,
%U A353484 0,4,5,19,6,8,3,20,0,21,12,2,7,2,6,22,2,0,13,23,3,5,14,9,4,24,2,3,8,10,15,6,1,25
%N A353484 a(1) = 0; and for n > 1, a(n) = A165560(n) + a(A064989(n)), where A165560 is the parity of arithmetic derivative, and A064989 shifts the prime factorization of its argument one step toward lower primes.
%C A353484 a(n) counts the number of the terms of A235991 encountered [including also n itself if the arithmetic derivative of n is odd] when repeatedly prime shifting n down to 1.
%H A353484 Antti Karttunen, <a href="/A353484/b353484.txt">Table of n, a(n) for n = 1..16384</a>
%H A353484 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A353484 For n >= 1, a(A000040(n)) = n, a(n^2) = 0.
%o A353484 (PARI)
%o A353484 A000265(n) = (n>>valuation(n,2));
%o A353484 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A353484 A064989(n) = { my(f=factor(A000265(n))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
%o A353484 A353484(n) = if(1==n, 0, (A003415(n)%2) + A353484(A064989(n)));
%Y A353484 Cf. A003415, A064989, A165560, A235991, A353485 (positions of zeros).
%Y A353484 Cf. also A292375, A319703, A343221, A344027, A353515.
%K A353484 nonn,look
%O A353484 1,3
%A A353484 _Antti Karttunen_, Apr 22 2022