A353333 Number of ways to write n as a product of the terms of A340784 larger than 1; a(1) = 1 by convention (an empty product).

1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 2, 1, 0, 1, 2, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0

Offset

1,16

Comments

Number of factorizations of n into factors k > 1 for which both A001222(k) and A056239(k) are even.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from indices in prime factorization

Formula

a(p) = 0 for all primes p.

a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.

Example

Of the eleven divisors of 220 larger than one, only [4, 10, 22, 55, 220] are in A340784, as both the number of their prime factors (with repetition, A001222), [2, 2, 2, 2, 4], and their integer pseudo logarithms (A056239), [2, 4, 6, 8, 10], are even. Using these factors gives the following possible factorizations: 220 = 22*10 = 55*4, therefore a(220) = 3.
Of the eight divisors of 256 larger than one, only [1, 4, 16, 64, 256] are in A340784. Using these factors, we obtain the following factorizations: 256 = 64*4 = 16*16 = 16*4*4 = 4*4*4*4, therefore a(256) = 5.
Of the 23 divisors of 792 larger than one, only [4, 9, 22, 36, 88, 198, 792] are in A340784. Using these factors gives the following possible factorizations: 792 = 198*4 = 88*9 = 36*22 = 22*4*9, therefore a(792) = 5.

Prog

(PARI)
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
A353331(n) = ((!(bigomega(n)%2)) && (!(A056239(n)%2)));
A353333(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1) && (d<=m) && A353331(d), s += A353333(n/d, d))); (s));

Crossrefs

Cf. A001222, A003961, A056239, A340784, A348717, A353331, A353334 [= a(n^2)].

Differs from A353303 for the first time at n=30, where a(30) = 0, while A353303(30) = 1.

Sequence in context: A331902 A333817 A270417 * A353303 A307666 A319995

Adjacent sequences: A353330 A353331 A353332 * A353334 A353335 A353336

Keyword

nonn

Author

Antti Karttunen, Apr 14 2022

Status

approved