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Number of factorizations of n into factors > 1 with length and greatest factor equal.
24

%I #11 Jun 19 2024 16:17:46

%S 0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,

%T 0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,

%U 0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1

%N Number of factorizations of n into factors > 1 with length and greatest factor equal.

%C I call these alt-balanced factorizations. Balanced factorizations are A340653. - _Gus Wiseman_, Jan 20 2021

%H Antti Karttunen, <a href="/A340599/b340599.txt">Table of n, a(n) for n = 1..100000</a>

%e The alt-balanced factorizations for n = 192, 1728, 3456, 9216:

%e 3*4*4*4 2*2*2*6*6*6 2*2*4*6*6*6 4*4*4*4*6*6

%e 2*2*2*2*2*6 2*2*3*4*6*6 2*3*4*4*6*6 2*2*2*2*2*6*6*8

%e 2*3*3*4*4*6 3*3*4*4*4*6 2*2*2*2*3*3*8*8

%e 2*2*2*2*3*3*3*8 2*2*2*2*3*4*6*8

%e 2*2*2*2*2*2*2*3*9 2*2*2*3*3*4*4*8

%e 2*2*2*2*2*2*2*8*9

%e 2*2*2*2*2*2*4*4*9

%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[n],Length[#]==Max[#]&]],{n,100}]

%o (PARI) A340599(n, m=n, e=0, mf=1) = if(1==n, mf==e, sumdiv(n, d, if((d>1)&&(d<=m), A340599(n/d, d, 1+e, max(d, mf))))); \\ _Antti Karttunen_, Jun 19 2024

%Y The co-balanced version is A340596.

%Y Positions of nonzero terms are A340597.

%Y The case of powers of two is A340611.

%Y Taking maximum Omega instead of maximum factor gives A340653.

%Y The cross-balanced version is A340654.

%Y The twice-balanced version is A340655.

%Y A001055 counts factorizations.

%Y A045778 counts strict factorizations.

%Y A316439 counts factorizations by product and length.

%Y Other balance-related sequences:

%Y - A010054 counts balanced strict partitions.

%Y - A047993 counts balanced partitions.

%Y - A098124 counts balanced compositions.

%Y - A106529 lists Heinz numbers of balanced partitions.

%Y - A340598 counts balanced set partitions.

%Y - A340600 counts unlabeled balanced multiset partitions.

%Y Cf. A006141, A117409, A320655, A320656, A324518, A339846, A339890, A340607.

%K nonn

%O 1,192

%A _Gus Wiseman_, Jan 20 2021

%E Data section extended up to a(120) and the secondary offset added by _Antti Karttunen_, Jun 19 2024