# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: id:a340654
Showing 1-1 of 1

%I A340654 #16 Jun 19 2024 16:17:50
%S A340654 1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,3,1,1,1,2,1,1,1,1,1,1,
%T A340654 1,5,1,1,1,3,1,1,1,2,2,1,1,4,1,2,1,2,1,3,1,3,1,1,1,3,1,1,2,1,1,1,1,2,
%U A340654 1,1,1,6,1,1,2,2,1,1,1,4,1,1,1,3,1,1,1,3,1,3,1,2,1,1,1,5,1,2,2,5,1,1,1,3,1
%N A340654 Number of cross-balanced factorizations of n.
%C A340654 We define a factorization of n into factors > 1 to be cross-balanced if either (1) it is empty or (2) the maximum image of A001222 over the factors is A001221(n).
%H A340654 Antti Karttunen, <a href="/A340654/b340654.txt">Table of n, a(n) for n = 1..65537</a>
%e A340654 The cross-balanced factorizations for n = 12, 24, 36, 72, 144, 240:
%e A340654   2*6   4*6     4*9     2*4*9     4*4*9       8*30
%e A340654   3*4   2*2*6   6*6     2*6*6     4*6*6       12*20
%e A340654         2*3*4   2*2*9   3*4*6     2*2*4*9     5*6*8
%e A340654                 2*3*6   2*2*2*9   2*2*6*6     2*4*30
%e A340654                 3*3*4   2*2*3*6   2*3*4*6     2*6*20
%e A340654                         2*3*3*4   3*3*4*4     2*8*15
%e A340654                                   2*2*2*2*9   3*4*20
%e A340654                                   2*2*2*3*6   3*8*10
%e A340654                                   2*2*3*3*4   4*5*12
%e A340654                                               2*10*12
%e A340654                                               2*3*5*8
%e A340654                                               2*2*2*30
%e A340654                                               2*2*3*20
%e A340654                                               2*2*5*12
%t A340654 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A340654 Table[Length[Select[facs[n],#=={}||PrimeNu[n]==Max[PrimeOmega/@#]&]],{n,100}]
%o A340654 (PARI) A340654(n, m=n, om=omega(n),mbo=0) = if(1==n,(mbo==om), sumdiv(n, d, if((d>1)&&(d<=m), A340654(n/d, d, om, max(mbo,bigomega(d)))))); \\ _Antti Karttunen_, Jun 19 2024
%Y A340654 Positions of terms > 1 are A126706.
%Y A340654 Positions of 1's are A303554.
%Y A340654 The co-balanced version is A340596.
%Y A340654 The version for unlabeled multiset partitions is A340651.
%Y A340654 The balanced version is A340653.
%Y A340654 The twice-balanced version is A340655.
%Y A340654 A001055 counts factorizations.
%Y A340654 A045778 counts strict factorizations.
%Y A340654 A316439 counts factorizations by product and length.
%Y A340654 A320655 counts factorizations into semiprimes.
%Y A340654 Other balance-related sequences:
%Y A340654 - A010054 counts balanced strict partitions.
%Y A340654 - A047993 counts balanced partitions.
%Y A340654 - A098124 counts balanced compositions.
%Y A340654 - A106529 lists Heinz numbers of balanced partitions.
%Y A340654 - A340597 have an alt-balanced factorization.
%Y A340654 - A340598 counts balanced set partitions.
%Y A340654 - A340599 counts alt-balanced factorizations.
%Y A340654 - A340652 counts unlabeled twice-balanced multiset partitions.
%Y A340654 - A340656 have no twice-balanced factorizations.
%Y A340654 - A340657 have a twice-balanced factorization.
%Y A340654 Cf. A003963, A117409, A303975, A320656, A324518, A339846, A339890, A340608.
%K A340654 nonn
%O A340654 1,12
%A A340654 _Gus Wiseman_, Jan 15 2021
%E A340654 Data section extended up to a(105) by _Antti Karttunen_, Jun 19 2024

# Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE