%I #7 Jul 22 2022 20:51:54
%S 1,2,6,9,15,49,35,27,45,98,63,105,171,117,81,135,245,343,273,549,189,
%T 1083,315,5618,741,686,507,513,351,243,405,7467,6419,5575,735,6859,
%U 1813,3231,1183,1197,3537,819,1647,567,945,2197,8397,3211,1715,3249,3367
%N 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 This is the position of first appearance of n in A355737.
%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.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Coprime_integers">Coprime integers</a>.
%e The terms together with their prime indices begin:
%e 1: {}
%e 2: {1}
%e 6: {1,2}
%e 9: {2,2}
%e 15: {2,3}
%e 49: {4,4}
%e 35: {3,4}
%e 27: {2,2,2}
%e 45: {2,2,3}
%e 98: {1,4,4}
%e 63: {2,2,4}
%e 105: {2,3,4}
%e 171: {2,2,8}
%e 117: {2,2,6}
%e 81: {2,2,2,2}
%e 135: {2,2,2,3}
%e For example, the choices for a(12) = 105 are:
%e (1,1,1) (1,3,2) (2,1,4)
%e (1,1,2) (1,3,4) (2,3,1)
%e (1,1,4) (2,1,1) (2,3,2)
%e (1,3,1) (2,1,2) (2,3,4)
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
%t az=Table[Length[Select[Tuples[Divisors/@primeMS[n]],GCD@@#==1&]],{n,100}];
%t Table[Position[az+1,k][[1,1]],{k,mnrm[az+1]}]
%Y Not requiring coprimality gives A355732, firsts of A355731.
%Y Positions of first appearances in A355737.
%Y A000005 counts divisors.
%Y A001221 counts distinct prime factors, with sum A001414.
%Y A001222 counts prime factors with multiplicity.
%Y A003963 multiplies together the prime indices of n.
%Y A056239 adds up prime indices, row sums of A112798.
%Y A120383 lists numbers divisible by all of their prime indices.
%Y A289508 gives GCD of prime indices.
%Y A289509 ranks relatively prime partitions, odd A302697, squarefree A302796.
%Y A324850 lists numbers divisible by the product of their prime indices.
%Y Cf. A000720, A007359, A051424, A076610, A302696, A302698, A355733, A355735, A355739, A355741, A355748.
%K nonn
%O 1,2
%A _Gus Wiseman_, Jul 21 2022