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Number of factorizations of n whose conjugate as an integer partition has no ones.
4

%I #10 May 03 2022 14:37:34

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

%C 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 Antti Karttunen, <a href="/A325045/b325045.txt">Table of n, a(n) for n = 1..65537</a>

%e The initial terms count the following factorizations:

%e 1: {}

%e 4: 2*2

%e 8: 2*2*2

%e 9: 3*3

%e 16: 2*2*2*2

%e 16: 4*4

%e 18: 2*3*3

%e 25: 5*5

%e 27: 3*3*3

%e 32: 2*2*2*2*2

%e 32: 2*4*4

%e 36: 2*2*3*3

%e 36: 6*6

%e 48: 3*4*4

%e 49: 7*7

%e 50: 2*5*5

%e 54: 2*3*3*3

%e 64: 2*2*2*2*2*2

%e 64: 2*2*4*4

%e 64: 4*4*4

%e 64: 8*8

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];

%t Table[Length[Select[facs[n],FreeQ[conj[#],1]&]],{n,1,100}]

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

%K nonn

%O 1,16

%A _Gus Wiseman_, Mar 27 2019

%E More terms from _Antti Karttunen_, May 03 2022