cp's OEIS Frontend

A355732 Least k such that there are exactly n ways to choose a sequence of divisors, one of each element of the multiset of prime indices of k (with multiplicity).

%I A355732 #7 Jul 22 2022 17:45:12
%S A355732 1,3,7,9,53,21,311,27,49,159,8161,63,38873,933,371,81,147,477,2177,
%T A355732 24483,189,2809,343,2799,1113,243,57127,16483,441,1431,6531,73449,
%U A355732 2597,567,96721,8427,1029,8397,3339,15239,729,49449,1323,19663,4293,2401,19593,7791
%N A355732 Least k such that there are exactly n ways to choose a sequence of divisors, one of each element of the multiset of prime indices of k (with multiplicity).
%C A355732 This is the position of first appearance of n in A355731.
%C A355732 Appears to be a subset of A353397.
%C A355732 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.
%e A355732 The terms together with their prime indices begin:
%e A355732       1: {}
%e A355732       3: {2}
%e A355732       7: {4}
%e A355732       9: {2,2}
%e A355732      53: {16}
%e A355732      21: {2,4}
%e A355732     311: {64}
%e A355732      27: {2,2,2}
%e A355732      49: {4,4}
%e A355732     159: {2,16}
%e A355732    8161: {1024}
%e A355732      63: {2,2,4}
%e A355732 For example, the choices for a(12) = 63 are:
%e A355732   (1,1,1)  (1,2,2)  (2,1,4)
%e A355732   (1,1,2)  (1,2,4)  (2,2,1)
%e A355732   (1,1,4)  (2,1,1)  (2,2,2)
%e A355732   (1,2,1)  (2,1,2)  (2,2,4)
%t A355732 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A355732 mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
%t A355732 az=Table[Times@@Length/@Divisors/@primeMS[n],{n,1000}];
%t A355732 Table[Position[az,k][[1,1]],{k,mnrm[az]}]
%Y A355732 Positions of first appearances in A355731.
%Y A355732 Counting distinct sequences after sorting: A355734, firsts of A355733.
%Y A355732 Requiring the result to be weakly increasing: A355736, firsts of A355735.
%Y A355732 Requiring the result to be relatively prime: A355738, firsts of A355737.
%Y A355732 A000005 counts divisors.
%Y A355732 A001414 adds up distinct prime divisors, counted by A001221.
%Y A355732 A003963 multiplies together the prime indices of n.
%Y A355732 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A355732 A120383 lists numbers divisible by all of their prime indices.
%Y A355732 A324850 lists numbers divisible by the product of their prime indices.
%Y A355732 Cf. A000720, A076610, A340606, A355739, A355740, A355741, A355742, A355744, A355746, A355747, A355748.
%K A355732 nonn
%O A355732 1,2
%A A355732 _Gus Wiseman_, Jul 21 2022