A353484 - OEIS

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

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OFFSET

1,3

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

FORMULA

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

PROG

(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)));

CROSSREFS