cp's OEIS Frontend

A348616 Number of ordered factorizations of n with adjacent equal factors.

%I A348616 #8 Dec 04 2021 12:37:13
%S A348616 0,0,0,1,0,0,0,1,1,0,0,2,0,0,0,4,0,2,0,2,0,0,0,6,1,0,1,2,0,0,0,9,0,0,
%T A348616 0,9,0,0,0,6,0,0,0,2,2,0,0,19,1,2,0,2,0,6,0,6,0,0,0,8,0,0,2,18,0,0,0,
%U A348616 2,0,0,0,31,0,0,2,2,0,0,0,19,4,0,0,8,0,0
%N A348616 Number of ordered factorizations of n with adjacent equal factors.
%C A348616 First differs from A348613 at a(24) = 6, A348613(24) = 8.
%C A348616 An ordered factorization of n is a finite sequence of positive integers > 1 with product n.
%F A348616 a(n) = A074206(n) - A348611(n).
%e A348616 The a(n) ordered factorizations with at least one pair of adjacent equal factors for n = 12, 24, 36, 60:
%e A348616    2*2*3    2*2*6      6*6        15*2*2
%e A348616    3*2*2    6*2*2      2*2*9      2*2*15
%e A348616             2*2*2*3    3*3*4      2*2*3*5
%e A348616             2*2*3*2    4*3*3      2*2*5*3
%e A348616             2*3*2*2    9*2*2      3*2*2*5
%e A348616             3*2*2*2    2*2*3*3    3*5*2*2
%e A348616                        2*3*3*2    5*2*2*3
%e A348616                        3*2*2*3    5*3*2*2
%e A348616                        3*3*2*2
%e A348616 See also examples in A348611.
%t A348616 ordfacs[n_]:=If[n<=1,{{}},Join@@Table[Prepend[#,d]&/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
%t A348616 antirunQ[y_]:=Length[y]==Length[Split[y]]
%t A348616 Table[Length[Select[ordfacs[n],!antirunQ[#]&]],{n,100}]
%Y A348616 Positions of 0's are A005117.
%Y A348616 Positions of 4's appear to be A030514.
%Y A348616 Positions of 2's appear to be A054753.
%Y A348616 Positions of 1's appear to be A168363.
%Y A348616 The additive version (compositions) is A261983, complement A003242.
%Y A348616 Factorizations with a permutation of this type are counted by A333487.
%Y A348616 Factorizations without a permutation of this type are counted by A335434.
%Y A348616 The complement is counted by A348611.
%Y A348616 As compositions these are ranked by A348612, complement A333489.
%Y A348616 Dominated by A348613 (non-alternating ordered factorizations).
%Y A348616 A001055 counts factorizations, strict A045778, ordered A074206.
%Y A348616 A335452 counts anti-run permutations of prime indices, complement A336107.
%Y A348616 A339846 counts even-length factorizations.
%Y A348616 A339890 counts odd-length factorizations.
%Y A348616 Cf. A001250, A025047, A122181, A138364, A347050, A347463, A347706, A348379, A348380, A348382, A348610.
%K A348616 nonn
%O A348616 1,12
%A A348616 _Gus Wiseman_, Nov 08 2021