OFFSET
1,6
COMMENTS
EXAMPLE
The a(n) ordered factorizations without adjacent equal factors for n = 1, 6, 12, 16, 24, 30, 32, 36 are:
() 6 12 16 24 30 32 36
2*3 2*6 2*8 3*8 5*6 4*8 4*9
3*2 3*4 8*2 4*6 6*5 8*4 9*4
4*3 2*4*2 6*4 10*3 16*2 12*3
6*2 8*3 15*2 2*16 18*2
2*3*2 12*2 2*15 2*8*2 2*18
2*12 3*10 4*2*4 3*12
2*3*4 2*3*5 2*3*6
2*4*3 2*5*3 2*6*3
2*6*2 3*2*5 2*9*2
3*2*4 3*5*2 3*2*6
3*4*2 5*2*3 3*4*3
4*2*3 5*3*2 3*6*2
4*3*2 6*2*3
6*3*2
2*3*2*3
3*2*3*2
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
MATHEMATICA
ordfacs[n_]:=If[n<=1, {{}}, Join@@Table[Prepend[#, d]&/@ordfacs[n/d], {d, Rest[Divisors[n]]}]];
antirunQ[y_]:=Length[y]==Length[Split[y]]
Table[Length[Select[ordfacs[n], antirunQ]], {n, 100}]
PROG
(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
CROSSREFS
Factorizations without a permutation of this type are counted by A333487.
Factorizations with a permutation of this type are counted by A335434.
The complement is counted by A348616.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A348613 counts non-alternating ordered factorizations.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 07 2021
STATUS
approved