A355732 - OEIS

%I #7 Jul 22 2022 17:45:12

%S 1,3,7,9,53,21,311,27,49,159,8161,63,38873,933,371,81,147,477,2177,

%T 24483,189,2809,343,2799,1113,243,57127,16483,441,1431,6531,73449,

%U 2597,567,96721,8427,1029,8397,3339,15239,729,49449,1323,19663,4293,2401,19593,7791

%N 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 This is the position of first appearance of n in A355731.

%C Appears to be a subset of A353397.

%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.

%e The terms together with their prime indices begin:

%e       1: {}

%e       3: {2}

%e       7: {4}

%e       9: {2,2}

%e      53: {16}

%e      21: {2,4}

%e     311: {64}

%e      27: {2,2,2}

%e      49: {4,4}

%e     159: {2,16}

%e    8161: {1024}

%e      63: {2,2,4}

%e For example, the choices for a(12) = 63 are:

%e   (1,1,1)  (1,2,2)  (2,1,4)

%e   (1,1,2)  (1,2,4)  (2,2,1)

%e   (1,1,4)  (2,1,1)  (2,2,2)

%e   (1,2,1)  (2,1,2)  (2,2,4)

%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[Times@@Length/@Divisors/@primeMS[n],{n,1000}];

%t Table[Position[az,k][[1,1]],{k,mnrm[az]}]

%Y Positions of first appearances in A355731.

%Y Counting distinct sequences after sorting: A355734, firsts of A355733.

%Y Requiring the result to be weakly increasing: A355736, firsts of A355735.

%Y Requiring the result to be relatively prime: A355738, firsts of A355737.

%Y A000005 counts divisors.

%Y A001414 adds up distinct prime divisors, counted by A001221.

%Y A003963 multiplies together the prime indices of n.

%Y A056239 adds up prime indices, row sums of A112798, counted by A001222.

%Y A120383 lists numbers divisible by all of their prime indices.

%Y A324850 lists numbers divisible by the product of their prime indices.

%Y Cf. A000720, A076610, A340606, A355739, A355740, A355741, A355742, A355744, A355746, A355747, A355748.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jul 21 2022