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

1,16

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.

Links

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

Example

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

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}]

Prog

(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

Crossrefs

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

Sequence in context: A307832 A037047 A118917 * A204293 A206479 A219484

Adjacent sequences: A325042 A325043 A325044 * A325046 A325047 A325048

Keyword

nonn

Author

Gus Wiseman, Mar 27 2019

Extensions

More terms from Antti Karttunen, May 03 2022

Status

approved