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Products of elements of A304711.
1

%I #7 Aug 06 2020 09:27:55

%S 1,2,4,6,8,10,12,14,15,16,18,20,22,24,26,28,30,32,33,34,35,36,38,40,

%T 44,45,46,48,50,51,52,54,55,56,58,60,62,64,66,68,69,70,72,74,75,76,77,

%U 80,82,84,85,86,88,90,92,93,94,95,96,98,99,100,102,104,106

%N Products of elements of A304711.

%C A304711 lists numbers whose distinct prime indices are pairwise coprime.

%C First differs from A304711 in having 84.

%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vSX9dPMGJhxB8rOknCGvOs6PiyhupdWNpqLsnphdgU6MEVqFBnWugAXidDhwHeKqZe_YnUqYeGOXsOk/pub">Sequences counting and encoding certain classes of multisets</a>

%e The sequence of terms together with their prime indices begins:

%e 1: {} 28: {1,1,4} 52: {1,1,6}

%e 2: {1} 30: {1,2,3} 54: {1,2,2,2}

%e 4: {1,1} 32: {1,1,1,1,1} 55: {3,5}

%e 6: {1,2} 33: {2,5} 56: {1,1,1,4}

%e 8: {1,1,1} 34: {1,7} 58: {1,10}

%e 10: {1,3} 35: {3,4} 60: {1,1,2,3}

%e 12: {1,1,2} 36: {1,1,2,2} 62: {1,11}

%e 14: {1,4} 38: {1,8} 64: {1,1,1,1,1,1}

%e 15: {2,3} 40: {1,1,1,3} 66: {1,2,5}

%e 16: {1,1,1,1} 44: {1,1,5} 68: {1,1,7}

%e 18: {1,2,2} 45: {2,2,3} 69: {2,9}

%e 20: {1,1,3} 46: {1,9} 70: {1,3,4}

%e 22: {1,5} 48: {1,1,1,1,2} 72: {1,1,1,2,2}

%e 24: {1,1,1,2} 50: {1,3,3} 74: {1,12}

%e 26: {1,6} 51: {2,7} 75: {2,3,3}

%t nn=100;

%t dat=Select[Range[nn],CoprimeQ@@PrimePi/@First/@FactorInteger[#]&];

%t facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];

%t Select[Range[nn],facsusing[dat,#]!={}&]

%Y A181818 is the version for superprimorials, with complement A336426.

%Y A336496 is the version for superfactorials, with complement A336497.

%Y A336620 is the complement.

%Y A000837 counts relatively prime partitions, with strict case A007360.

%Y A001055 counts factorizations.

%Y A302696 lists numbers with coprime prime indices.

%Y A304711 lists numbers with coprime distinct prime indices.

%Y Cf. A001221, A007360, A007916, A056239, A112798, A302569, A327516, A333228, A335238, A335239, A335240, A336424, A336497, A336736.

%K nonn

%O 1,2

%A _Gus Wiseman_, Aug 02 2020