cp's OEIS Frontend

A348611 Number of ordered factorizations of n with no adjacent equal factors.

%I A348611 #17 Nov 12 2021 22:31:21
%S A348611 1,1,1,1,1,3,1,3,1,3,1,6,1,3,3,4,1,6,1,6,3,3,1,14,1,3,3,6,1,13,1,7,3,
%T A348611 3,3,17,1,3,3,14,1,13,1,6,6,3,1,29,1,6,3,6,1,14,3,14,3,3,1,36,1,3,6,
%U A348611 14,3,13,1,6,3,13,1,45,1,3,6,6,3,13,1,29,4,3
%N A348611 Number of ordered factorizations of n with no adjacent equal factors.
%C A348611 First differs from A348610 at a(24) = 14, A348610(24) = 12.
%C A348611 An ordered factorization of n is a finite sequence of positive integers > 1 with product n.
%C A348611 In analogy with Carlitz compositions, these may be called Carlitz ordered factorizations.
%F A348611 a(n) = A074206(n) - A348616(n).
%e A348611 The a(n) ordered factorizations without adjacent equal factors for n = 1, 6, 12, 16, 24, 30, 32, 36 are:
%e A348611   ()   6     12      16      24      30      32      36
%e A348611        2*3   2*6     2*8     3*8     5*6     4*8     4*9
%e A348611        3*2   3*4     8*2     4*6     6*5     8*4     9*4
%e A348611              4*3     2*4*2   6*4     10*3    16*2    12*3
%e A348611              6*2             8*3     15*2    2*16    18*2
%e A348611              2*3*2           12*2    2*15    2*8*2   2*18
%e A348611                              2*12    3*10    4*2*4   3*12
%e A348611                              2*3*4   2*3*5           2*3*6
%e A348611                              2*4*3   2*5*3           2*6*3
%e A348611                              2*6*2   3*2*5           2*9*2
%e A348611                              3*2*4   3*5*2           3*2*6
%e A348611                              3*4*2   5*2*3           3*4*3
%e A348611                              4*2*3   5*3*2           3*6*2
%e A348611                              4*3*2                   6*2*3
%e A348611                                                      6*3*2
%e A348611                                                      2*3*2*3
%e A348611                                                      3*2*3*2
%e A348611 Thus, of total A074206(12) = 8 ordered factorizations of 12, only factorizations 2*2*3 and 3*2*2 (see A348616) are not included in this count, therefore a(12) = 6. - _Antti Karttunen_, Nov 12 2021
%t A348611 ordfacs[n_]:=If[n<=1,{{}},Join@@Table[Prepend[#,d]&/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
%t A348611 antirunQ[y_]:=Length[y]==Length[Split[y]]
%t A348611 Table[Length[Select[ordfacs[n],antirunQ]],{n,100}]
%o A348611 (PARI) A348611(n, e=0) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d!=e), s += A348611(n/d, d))); (s)); \\ _Antti Karttunen_, Nov 12 2021
%Y A348611 The additive version (compositions) is A003242, complement A261983.
%Y A348611 The additive alternating version is A025047, ranked by A345167.
%Y A348611 Factorizations without a permutation of this type are counted by A333487.
%Y A348611 As compositions these are ranked by A333489, complement A348612.
%Y A348611 Factorizations with a permutation of this type are counted by A335434.
%Y A348611 The non-alternating additive version is A345195, ranked by A345169.
%Y A348611 The alternating case is A348610, which is dominated at positions A122181.
%Y A348611 The complement is counted by A348616.
%Y A348611 A001055 counts factorizations, strict A045778, ordered A074206.
%Y A348611 A325534 counts separable partitions, ranked by A335433.
%Y A348611 A335452 counts anti-run permutations of prime indices, complement A336107.
%Y A348611 A339846 counts even-length factorizations.
%Y A348611 A339890 counts odd-length factorizations.
%Y A348611 A348613 counts non-alternating ordered factorizations.
%Y A348611 Cf. A001250, A138364, A336103, A347050, A347438, A347463, A347466, A347706, A348379, A348383.
%K A348611 nonn
%O A348611 1,6
%A A348611 _Gus Wiseman_, Nov 07 2021