A353645 a(1) = 1, and for n > 1, number of ways to write the square of n as a product of at least two factors, the largest two of which are equal.

1, 1, 1, 2, 1, 2, 1, 4, 2, 3, 1, 5, 1, 3, 2, 7, 1, 5, 1, 6, 2, 3, 1, 10, 2, 3, 4, 6, 1, 8, 1, 12, 3, 3, 2, 15, 1, 3, 3, 12, 1, 9, 1, 7, 5, 3, 1, 21, 2, 6, 3, 7, 1, 13, 2, 12, 3, 3, 1, 21, 1, 3, 5, 21, 2, 11, 1, 8, 3, 7, 1, 35, 1, 3, 5, 8, 2, 12, 1, 23, 7, 3, 1, 25, 2, 3, 3, 15, 1, 21, 2, 8, 3, 3, 2, 43, 1, 6, 6, 16

Offset: 1

Author

Antti Karttunen, May 03 2022

Keywords

• nonn

Examples

			For n=6, 6^2 = 36 can be factorized in two ways so that two largest factors are equal, as 2*2*3*3 = 6*6, therefore a(6) = 2. See also the examples in A325045.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16415

Crossrefs

Cf. A000290, A325045.

Programs

PARI

A325045(n, m=n, facs=List([])) = if(1==n, (0==#facs || (#facs>=2 && facs[1]==facs[2])), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A325045(n/d, d, newfacs))); (s));
A353645(n) = A325045(n^2);

Formula

a(n) = A325045(A000290(n)).