A335427 a(1) = 0; for k >= 2, a(prime(k)) = 0, a(k^2) = 2 * a(k); otherwise a(n) = a(A334870(n)) + 1.

0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 4, 0, 1, 2, 4, 0, 1, 0, 6, 2, 1, 0, 5, 0, 1, 2, 10, 0, 3, 0, 5, 2, 1, 4, 2, 0, 1, 2, 7, 0, 3, 0, 18, 4, 1, 0, 6, 0, 1, 2, 34, 0, 3, 4, 11, 2, 1, 0, 8, 0, 1, 8, 6, 4, 3, 0, 66, 2, 5, 0, 3, 0, 1, 2, 130, 8, 3, 0, 8, 0, 1, 0, 12, 4, 1, 2, 19, 0, 5, 8, 258, 2, 1, 4, 7, 0, 1, 16, 2, 0, 3, 0, 35, 6

Offset

1,4

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Index entries for sequences computed from indices in prime factorization

Formula

Alternative definition: (Start)

a(1) = 0, a(2) = 1; otherwise for n = k * m^2, k squarefree:

if m = 1, a(n) = A048675(A052126(k));

if m > 1, a(n) = A048675(k) + 2 * a(m).

(End)

For n = 4 * A122132(k), a(n) = A048675(n).

More generally, a(n) = A048675(n) if and only if n is in A335738.

a(n) = A335426(A225546(n)).

a(A003961(2k+1)) = 2 * a(2k+1).

If n is in A036554, a(n) = a(n/2) + 1; otherwise for n <> 3, a(n) = 2 * a(A019565(k/2) * m^2) - a(m^2), where n = A019565(k) * m^2.

Prog

(PARI)
A334870(n) = if(issquare(n), sqrtint(n), my(c=core(n), m=n); forprime(p=2, , if(!(c % p), m/=p; break, m*=p)); (m));
A335427(n) = if(n<=2, n-1, if(isprime(n), 0, if(issquare(n), 2*A335427(sqrtint(n)), 1+A335427(A334870(n)))));
(PARI)
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A052126(n) = if(1==n, n, (n/vecmax(factor(n)[, 1])));
A335427(n) = if(n<=2, n-1, if(issquarefree(n), A048675(A052126(n)), my(k=core(n)); A048675(k) + 2*A335427(sqrtint(n/k))));

Crossrefs

A052126, A225546, A334870, A335426 are used in formulas defining this sequence.

Related fully additive sequence: A048675.

Cf. A062090 (indices of zeros), A003159 (indices of even values), A036554 (indices of odd values).

Cf. A005117, A122132, A334872, A335738.

A003961, A019565 are used to express relationship between terms of this sequence.

Sequence in context: A320439 A321252 A225345 * A083280 A060689 A053119

Adjacent sequences: A335424 A335425 A335426 * A335428 A335429 A335430

Keyword

nonn

Author

Antti Karttunen and Peter Munn, Jun 15 2020

Status

approved