A335426 -id:A335426 - cp's OEIS frontend

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

Antti Karttunen and Peter Munn, Jun 15 2020

nonn

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

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.

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)))));

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))));

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.

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula