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

Author

Antti Karttunen, Apr 22 2022

Keywords

• nonn
• look

Comments

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.

Links

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

Crossrefs

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

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

Programs

PARI

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)));

Formula

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