A353415 - OEIS

%I #8 Apr 19 2022 22:46:00

%S 1,1,1,1,1,3,1,3,1,2,1,6,1,3,3,5,1,6,1,5,2,2,1,14,1,3,3,6,1,12,1,6,3,

%T 2,3,20,1,3,2,10,1,12,1,5,6,2,1,27,1,5,3,6,1,14,2,14,2,3,1,32,1,2,5,

%U 11,3,12,1,5,3,12,1,43,1,3,6,6,3,12,1,17,5,2,1,38,2,3,2,10,1,38,2,5,3,2,3,46,1,6,6

%N Number of ways to write the square of n as a product of the terms of A353355 larger than 1; a(1) = 1 by convention (an empty product).

%C Number of factorizations of n^2 into factors k > 1 for which A353354(k) = 0.

%H Antti Karttunen, <a href="/A353415/b353415.txt">Table of n, a(n) for n = 1..16384</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%F a(n) = A353414(A000290(n)).

%F a(p) = 1 for all primes p.

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

%o (PARI)

%o A332823(n) = { my(f = factor(n),u=(sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3); if(2==u,-1,u); };

%o A353354(n) = sumdiv(n,d,A332823(d));

%o A353380(n) = (0==A353354(n));

%o A353414(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&A353380(d), s += A353414(n/d, d))); (s));

%o A353415(n) = A353414(n^2);

%Y Cf. A000290, A003961, A048675, A332823, A348717, A353352, A353354, A353355, A353380, A353414.

%K nonn

%O 1,6

%A _Antti Karttunen_, Apr 19 2022