A353485 -id:A353485 - 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.

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, 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, 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

Offset: 1

Antti Karttunen, Apr 22 2022

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.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A003415, A064989, A165560, A235991, A353485 (positions of zeros).

Cf. also A292375, A319703, A343221, A344027, A353515.

A000265(n) = (n>>valuation(n,2));
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A064989(n) = { my(f=factor(A000265(n))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A353484(n) = if(1==n, 0, (A003415(n)%2) + A353484(A064989(n)));

For n >= 1, a(A000040(n)) = n, a(n^2) = 0.

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.

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