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

%I A325045 #10 May 03 2022 14:37:34
%S A325045 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,
%T A325045 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,
%U A325045 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
%N A325045 Number of factorizations of n whose conjugate as an integer partition has no ones.
%C A325045 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.
%H A325045 Antti Karttunen, <a href="/A325045/b325045.txt">Table of n, a(n) for n = 1..65537</a>
%e A325045 The initial terms count the following factorizations:
%e A325045     1: {}
%e A325045     4: 2*2
%e A325045     8: 2*2*2
%e A325045     9: 3*3
%e A325045    16: 2*2*2*2
%e A325045    16: 4*4
%e A325045    18: 2*3*3
%e A325045    25: 5*5
%e A325045    27: 3*3*3
%e A325045    32: 2*2*2*2*2
%e A325045    32: 2*4*4
%e A325045    36: 2*2*3*3
%e A325045    36: 6*6
%e A325045    48: 3*4*4
%e A325045    49: 7*7
%e A325045    50: 2*5*5
%e A325045    54: 2*3*3*3
%e A325045    64: 2*2*2*2*2*2
%e A325045    64: 2*2*4*4
%e A325045    64: 4*4*4
%e A325045    64: 8*8
%t A325045 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A325045 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A325045 Table[Length[Select[facs[n],FreeQ[conj[#],1]&]],{n,1,100}]
%o A325045 (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
%Y A325045 Cf. A001055, A001222, A002865, A096276, A114324, A122111, A318950, A319005, A319916, A320322, A321648, A325039, A353645 [= a(n^2)].
%K A325045 nonn
%O A325045 1,16
%A A325045 _Gus Wiseman_, Mar 27 2019
%E A325045 More terms from _Antti Karttunen_, May 03 2022