A325045 Number of factorizations of n whose conjugate as an integer partition has no ones.

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0

Offset: 1

Author

Gus Wiseman, Mar 27 2019

Keywords

• nonn

Comments

After a(1) = 1, a(n) is the number of factorizations of n with at least two factors, the largest two of which are equal.

Examples

			The initial terms count the following factorizations:
    1: {}
    4: 2*2
    8: 2*2*2
    9: 3*3
   16: 2*2*2*2
   16: 4*4
   18: 2*3*3
   25: 5*5
   27: 3*3*3
   32: 2*2*2*2*2
   32: 2*4*4
   36: 2*2*3*3
   36: 6*6
   48: 3*4*4
   49: 7*7
   50: 2*5*5
   54: 2*3*3*3
   64: 2*2*2*2*2*2
   64: 2*2*4*4
   64: 4*4*4
   64: 8*8
		

Links

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

Crossrefs

Cf. A001055, A001222, A002865, A096276, A114324, A122111, A318950, A319005, A319916, A320322, A321648, A325039, A353645 [= a(n^2)].

Programs

Mathematica

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[facs[n],FreeQ[conj[#],1]&]],{n,1,100}]

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)); \\ Antti Karttunen, May 03 2022

Extensions

More terms from Antti Karttunen, May 03 2022