A336423 - OEIS

%I #20 May 08 2021 08:30:23

%S 1,1,1,2,1,0,1,4,2,0,1,5,1,0,0,8,1,5,1,5,0,0,1,14,2,0,4,5,1,0,1,16,0,

%T 0,0,0,1,0,0,14,1,0,1,5,5,0,1,36,2,5,0,5,1,14,0,14,0,0,1,0,1,0,5,32,0,

%U 0,1,5,0,0,1,35,1,0,5,5,0,0,1,36,8,0,1,0

%N Number of strict chains of divisors from n to 1 using terms of A130091 (numbers with distinct prime multiplicities).

%C A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

%e The a(n) chains for n = 4, 8, 12, 16, 24, 32:

%e   4/1    8/1      12/1      16/1        24/1         32/1

%e   4/2/1  8/2/1    12/2/1    16/2/1      24/2/1       32/2/1

%e          8/4/1    12/3/1    16/4/1      24/3/1       32/4/1

%e          8/4/2/1  12/4/1    16/8/1      24/4/1       32/8/1

%e                   12/4/2/1  16/4/2/1    24/8/1       32/16/1

%e                             16/8/2/1    24/12/1      32/4/2/1

%e                             16/8/4/1    24/4/2/1     32/8/2/1

%e                             16/8/4/2/1  24/8/2/1     32/8/4/1

%e                                         24/8/4/1     32/16/2/1

%e                                         24/12/2/1    32/16/4/1

%e                                         24/12/3/1    32/16/8/1

%e                                         24/12/4/1    32/8/4/2/1

%e                                         24/8/4/2/1   32/16/4/2/1

%e                                         24/12/4/2/1  32/16/8/2/1

%e                                                      32/16/8/4/1

%e                                                      32/16/8/4/2/1

%t strchns[n_]:=If[n==1,1,If[!UnsameQ@@Last/@FactorInteger[n],0,Sum[strchns[d],{d,Select[Most[Divisors[n]],UnsameQ@@Last/@FactorInteger[#]&]}]]];

%t Table[strchns[n],{n,100}]

%Y A336569 is the maximal case.

%Y A336571 does not require n itself to have distinct prime multiplicities.

%Y A000005 counts divisors.

%Y A007425 counts divisors of divisors.

%Y A074206 counts strict chains of divisors from n to 1.

%Y A130091 lists numbers with distinct prime multiplicities.

%Y A181796 counts divisors with distinct prime multiplicities.

%Y A253249 counts nonempty strict chains of divisors.

%Y A327498 gives the maximum divisor with distinct prime multiplicities.

%Y A336422 counts divisible pairs of divisors, both in A130091.

%Y A336424 counts factorizations using A130091.

%Y A336500 counts divisors of n in A130091 with quotient also in A130091.

%Y A337256 counts strict chains of divisors.

%Y Cf. A001055, A002033, A005117, A032741, A067824, A071625, A118914, A124010, A167865, A336568, A336570, A336941.

%K nonn

%O 1,4

%A _Gus Wiseman_, Jul 27 2020