|
%I #15 Mar 16 2023 17:28:50
%S 1,1,1,3,1,2,1,3,2,2,1,4,1,2,2,6,1,3,1,4,2,2,1,6,2,2,2,4,1,4,1,7,2,2,
%T 2,7,1,2,2,6,1,4,1,4,3,2,1,10,2,3,2,4,1,4,2,6,2,2,1,8,1,2,3,12,2,4,1,
%U 4,2,4,1,10,1,2,3,4,2,4,1,10,3,2,1,8,2,2
%N Number of factorizations of 4n into an even number of even factors > 1.
%H Robert Israel, <a href="/A340786/b340786.txt">Table of n, a(n) for n = 1..10000</a>
%e The a(n) factorizations for n = 6, 12, 24, 36, 60, 80, 500:
%e 4*6 6*8 2*48 2*72 4*60 4*80 40*50
%e 2*12 2*24 4*24 4*36 6*40 8*40 4*500
%e 4*12 6*16 6*24 8*30 10*32 8*250
%e 2*2*2*6 8*12 8*18 10*24 16*20 10*200
%e 2*2*4*6 12*12 12*20 2*160 20*100
%e 2*2*2*12 2*2*6*6 2*120 2*2*2*40 2*1000
%e 2*2*2*18 2*2*2*30 2*2*4*20 2*2*10*50
%e 2*2*6*10 2*2*8*10 2*2*2*250
%e 2*4*4*10 2*10*10*10
%e 2*2*2*2*2*10
%p g:= proc(n, m, p)
%p option remember;
%p local F,r,x,i;
%p # number of factorizations of n into even factors > m with number of factors == p (mod 2)
%p if n = 1 then if p = 0 then return 1 else return 0 fi fi;
%p if m > n or n::odd then return 0 fi;
%p F:= sort(convert(select(t -> t > m and t::even, numtheory:-divisors(n)),list));
%p r:= 0;
%p for x in F do
%p for i from 1 while n mod x^i = 0 do
%p r:= r + procname(n/x^i, x, (p+i) mod 2)
%p od od;
%p r
%p end proc:
%p f:= n -> g(4*n, 1, 0):
%p map(f, [$1..100]); # _Robert Israel_, Mar 16 2023
%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t Table[Length[Select[facs[4n],EvenQ[Length[#]]&&Select[#,OddQ]=={}&]],{n,100}]
%o (PARI)
%o A340786aux(n, m=n, p=0) = if(1==n, (0==p), my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&!(d%2), s += A340786aux(n/d, d, 1-p))); (s));
%o A340786(n) = A340786aux(4*n); \\ _Antti Karttunen_, Apr 14 2022
%Y Note: A-numbers of Heinz-number sequences are in parentheses below.
%Y Positions of ones are 1 and A000040, or A008578.
%Y A version for partitions is A027187 (A028260).
%Y Allowing odd length gives A108501 (bisection of A340785).
%Y Allowing odd factors gives A339846.
%Y An odd version is A340102.
%Y - Factorizations -
%Y A001055 counts factorizations, with strict case A045778.
%Y A316439 counts factorizations by product and length.
%Y A340101 counts factorizations into odd factors.
%Y A340653 counts balanced factorizations.
%Y A340831/A340832 count factorizations with odd maximum/minimum.
%Y - Even -
%Y A027187 counts partitions of even maximum (A244990).
%Y A058696 counts partitions of even numbers (A300061).
%Y A067661 counts strict partitions of even length (A030229).
%Y A236913 counts partitions of even length and sum (A340784).
%Y A340601 counts partitions of even rank (A340602).
%Y Cf. A001147, A001222, A001749, A035363, A050320, A066207, A066208, A160786, A174725, A320655, A320656, A339890, A340851.
%K nonn
%O 1,4
%A _Gus Wiseman_, Jan 31 2021
|
