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A340599 Number of factorizations of n into factors > 1 with length and greatest factor equal.

%I A340599 #11 Jun 19 2024 16:17:46
%S A340599 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 A340599 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 A340599 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 A340599 Number of factorizations of n into factors > 1 with length and greatest factor equal.
%C A340599 I call these alt-balanced factorizations. Balanced factorizations are A340653. - _Gus Wiseman_, Jan 20 2021
%H A340599 Antti Karttunen, <a href="/A340599/b340599.txt">Table of n, a(n) for n = 1..100000</a>
%e A340599 The alt-balanced factorizations for n = 192, 1728, 3456, 9216:
%e A340599   3*4*4*4       2*2*2*6*6*6   2*2*4*6*6*6         4*4*4*4*6*6
%e A340599   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 A340599                 2*3*3*4*4*6   3*3*4*4*4*6         2*2*2*2*3*3*8*8
%e A340599                               2*2*2*2*3*3*3*8     2*2*2*2*3*4*6*8
%e A340599                               2*2*2*2*2*2*2*3*9   2*2*2*3*3*4*4*8
%e A340599                                                   2*2*2*2*2*2*2*8*9
%e A340599                                                   2*2*2*2*2*2*4*4*9
%t A340599 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A340599 Table[Length[Select[facs[n],Length[#]==Max[#]&]],{n,100}]
%o A340599 (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 A340599 The co-balanced version is A340596.
%Y A340599 Positions of nonzero terms are A340597.
%Y A340599 The case of powers of two is A340611.
%Y A340599 Taking maximum Omega instead of maximum factor gives A340653.
%Y A340599 The cross-balanced version is A340654.
%Y A340599 The twice-balanced version is A340655.
%Y A340599 A001055 counts factorizations.
%Y A340599 A045778 counts strict factorizations.
%Y A340599 A316439 counts factorizations by product and length.
%Y A340599 Other balance-related sequences:
%Y A340599 - A010054 counts balanced strict partitions.
%Y A340599 - A047993 counts balanced partitions.
%Y A340599 - A098124 counts balanced compositions.
%Y A340599 - A106529 lists Heinz numbers of balanced partitions.
%Y A340599 - A340598 counts balanced set partitions.
%Y A340599 - A340600 counts unlabeled balanced multiset partitions.
%Y A340599 Cf. A006141, A117409, A320655, A320656, A324518, A339846, A339890, A340607.
%K A340599 nonn
%O A340599 1,192
%A A340599 _Gus Wiseman_, Jan 20 2021
%E A340599 Data section extended up to a(120) and the secondary offset added by _Antti Karttunen_, Jun 19 2024