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A354911 Number of factorizations of n into relatively prime prime-powers.

%I A354911 #19 Jul 28 2022 21:15:24
%S A354911 1,0,0,0,0,1,0,0,0,1,0,2,0,1,1,0,0,2,0,2,1,1,0,3,0,1,0,2,0,1,0,0,1,1,
%T A354911 1,4,0,1,1,3,0,1,0,2,2,1,0,5,0,2,1,2,0,3,1,3,1,1,0,2,0,1,2,0,1,1,0,2,
%U A354911 1,1,0,6,0,1,2,2,1,1,0,5,0,1,0,2,1,1,1
%N A354911 Number of factorizations of n into relatively prime prime-powers.
%H A354911 Wikipedia, <a href="https://en.wikipedia.org/wiki/Coprime_integers">Coprime integers</a>.
%F A354911 a(n) = A000688(n) if n is nonprime, otherwise a(n) = 0.
%e A354911 The a(n) factorizations for n = 6, 12, 24, 36, 48, 72, 96:
%e A354911   2*3  3*4    3*8      4*9      3*16       8*9        3*32
%e A354911        2*2*3  2*3*4    2*2*9    2*3*8      2*4*9      3*4*8
%e A354911               2*2*2*3  3*3*4    3*4*4      3*3*8      2*3*16
%e A354911                        2*2*3*3  2*2*3*4    2*2*2*9    2*2*3*8
%e A354911                                 2*2*2*2*3  2*3*3*4    2*3*4*4
%e A354911                                            2*2*2*3*3  2*2*2*3*4
%e A354911                                                       2*2*2*2*2*3
%t A354911 ufacs[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[ufacs[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
%t A354911 Table[Length[Select[ufacs[Select[Divisors[n],PrimePowerQ[#]&],n],GCD@@#<=1&]],{n,100}]
%Y A354911 This is the relatively prime case of A000688, partitions A023894.
%Y A354911 Positions of 0's are A246655 (A000961 includes 1).
%Y A354911 For strict instead of relatively prime we have A050361, partitions A054685.
%Y A354911 Positions of 1's are A000469 (A120944 excludes 1).
%Y A354911 For pairwise coprime instead of relatively prime we have A143731.
%Y A354911 The version for partitions instead of factorizations is A356067.
%Y A354911 A000005 counts divisors.
%Y A354911 A001055 counts factorizations.
%Y A354911 A001221 counts distinct prime divisors, with sum A001414.
%Y A354911 A001222 counts prime-power divisors.
%Y A354911 A289509 lists numbers whose prime indices are relatively prime.
%Y A354911 A295935 counts twice-factorizations with constant blocks (type PPR).
%Y A354911 A355743 lists numbers with prime-power prime indices, squarefree A356065.
%Y A354911 Cf. A000837, A023893, A076610, A085970, A106244, A279784, A318721, A355737, A355742, A356064, A356066.
%K A354911 nonn
%O A354911 1,12
%A A354911 _Gus Wiseman_, Jul 25 2022