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A355738 Least k such that there are exactly n ways to choose a sequence of divisors, one of each prime index of k (with multiplicity), such that the result has no common divisor > 1.

%I A355738 #7 Jul 22 2022 20:51:54
%S A355738 1,2,6,9,15,49,35,27,45,98,63,105,171,117,81,135,245,343,273,549,189,
%T A355738 1083,315,5618,741,686,507,513,351,243,405,7467,6419,5575,735,6859,
%U A355738 1813,3231,1183,1197,3537,819,1647,567,945,2197,8397,3211,1715,3249,3367
%N A355738 Least k such that there are exactly n ways to choose a sequence of divisors, one of each prime index of k (with multiplicity), such that the result has no common divisor > 1.
%C A355738 This is the position of first appearance of n in A355737.
%C A355738 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.
%H A355738 Wikipedia, <a href="https://en.wikipedia.org/wiki/Coprime_integers">Coprime integers</a>.
%e A355738 The terms together with their prime indices begin:
%e A355738      1: {}
%e A355738      2: {1}
%e A355738      6: {1,2}
%e A355738      9: {2,2}
%e A355738     15: {2,3}
%e A355738     49: {4,4}
%e A355738     35: {3,4}
%e A355738     27: {2,2,2}
%e A355738     45: {2,2,3}
%e A355738     98: {1,4,4}
%e A355738     63: {2,2,4}
%e A355738    105: {2,3,4}
%e A355738    171: {2,2,8}
%e A355738    117: {2,2,6}
%e A355738     81: {2,2,2,2}
%e A355738    135: {2,2,2,3}
%e A355738 For example, the choices for a(12) = 105 are:
%e A355738   (1,1,1)  (1,3,2)  (2,1,4)
%e A355738   (1,1,2)  (1,3,4)  (2,3,1)
%e A355738   (1,1,4)  (2,1,1)  (2,3,2)
%e A355738   (1,3,1)  (2,1,2)  (2,3,4)
%t A355738 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A355738 mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
%t A355738 az=Table[Length[Select[Tuples[Divisors/@primeMS[n]],GCD@@#==1&]],{n,100}];
%t A355738 Table[Position[az+1,k][[1,1]],{k,mnrm[az+1]}]
%Y A355738 Not requiring coprimality gives A355732, firsts of A355731.
%Y A355738 Positions of first appearances in A355737.
%Y A355738 A000005 counts divisors.
%Y A355738 A001221 counts distinct prime factors, with sum A001414.
%Y A355738 A001222 counts prime factors with multiplicity.
%Y A355738 A003963 multiplies together the prime indices of n.
%Y A355738 A056239 adds up prime indices, row sums of A112798.
%Y A355738 A120383 lists numbers divisible by all of their prime indices.
%Y A355738 A289508 gives GCD of prime indices.
%Y A355738 A289509 ranks relatively prime partitions, odd A302697, squarefree A302796.
%Y A355738 A324850 lists numbers divisible by the product of their prime indices.
%Y A355738 Cf. A000720, A007359, A051424, A076610, A302696, A302698, A355733, A355735, A355739, A355741, A355748.
%K A355738 nonn
%O A355738 1,2
%A A355738 _Gus Wiseman_, Jul 21 2022