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A330998 Sorted list containing the least number whose inverse prime shadow (A181821) has each possible nonzero number of factorizations into factors > 1. 9

%I #8 Jan 08 2020 09:45:57

%S 1,3,5,6,7,9,10,12,14,15,16,18,19,20,21,22,23,24,25,27,28,29,30,31,33,

%T 34,35,36,37,38,40,41,42,43,44,45,46,47,49,51,52,53,54,56,57,58,59,60,

%U 61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79

%N Sorted list containing the least number whose inverse prime shadow (A181821) has each possible nonzero number of factorizations into factors > 1.

%C This is the sorted list of positions of first appearances in A318284 of each element of the range A045782.

%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The inverse prime shadow of n is the least number whose prime exponents are the prime indices of n.

%e Factorizations of the inverse prime shadows of the initial terms:

%e 4 8 12 16 36 24 60 48

%e 2*2 2*4 2*6 2*8 4*9 3*8 2*30 6*8

%e 2*2*2 3*4 4*4 6*6 4*6 3*20 2*24

%e 2*2*3 2*2*4 2*18 2*12 4*15 3*16

%e 2*2*2*2 3*12 2*2*6 5*12 4*12

%e 2*2*9 2*3*4 6*10 2*3*8

%e 2*3*6 2*2*2*3 2*5*6 2*4*6

%e 3*3*4 3*4*5 3*4*4

%e 2*2*3*3 2*2*15 2*2*12

%e 2*3*10 2*2*2*6

%e 2*2*3*5 2*2*3*4

%e 2*2*2*2*3

%e The corresponding multiset partitions:

%e {11} {111} {112} {1111} {1122} {1112}

%e {1}{1} {1}{11} {1}{12} {1}{111} {1}{122} {1}{112}

%e {1}{1}{1} {2}{11} {11}{11} {11}{22} {11}{12}

%e {1}{1}{2} {1}{1}{11} {12}{12} {2}{111}

%e {1}{1}{1}{1} {2}{112} {1}{1}{12}

%e {1}{1}{22} {1}{2}{11}

%e {1}{2}{12} {1}{1}{1}{2}

%e {2}{2}{11}

%e {1}{1}{2}{2}

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];

%t nds=Table[Length[facs[Times@@Prime/@nrmptn[n]]],{n,50}];

%t Table[Position[nds,i][[1,1]],{i,First/@Gather[nds]}]

%Y Taking n instead of the inverse prime shadow of n gives A330972.

%Y Factorizations are A001055, with image A045782, with complement A330976.

%Y Factorizations of inverse prime shadows are A318284.

%Y Cf. A025487, A033833, A045778, A045783, A181819, A181821, A318286, A325238, A330973, A330989, A330990, A330993, A330997.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jan 07 2020

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Last modified March 29 04:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)