# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a354911 Showing 1-1 of 1 %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, Coprime integers. %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 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE