login
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).
48
1, 3, 7, 9, 53, 21, 311, 27, 49, 159, 8161, 63, 38873, 933, 371, 81, 147, 477, 2177, 24483, 189, 2809, 343, 2799, 1113, 243, 57127, 16483, 441, 1431, 6531, 73449, 2597, 567, 96721, 8427, 1029, 8397, 3339, 15239, 729, 49449, 1323, 19663, 4293, 2401, 19593, 7791
OFFSET
1,2
COMMENTS
This is the position of first appearance of n in A355731.
Appears to be a subset of A353397.
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.
EXAMPLE
The terms together with their prime indices begin:
1: {}
3: {2}
7: {4}
9: {2,2}
53: {16}
21: {2,4}
311: {64}
27: {2,2,2}
49: {4,4}
159: {2,16}
8161: {1024}
63: {2,2,4}
For example, the choices for a(12) = 63 are:
(1,1,1) (1,2,2) (2,1,4)
(1,1,2) (1,2,4) (2,2,1)
(1,1,4) (2,1,1) (2,2,2)
(1,2,1) (2,1,2) (2,2,4)
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
mnrm[s_]:=If[Min@@s==1, mnrm[DeleteCases[s-1, 0]]+1, 0];
az=Table[Times@@Length/@Divisors/@primeMS[n], {n, 1000}];
Table[Position[az, k][[1, 1]], {k, mnrm[az]}]
CROSSREFS
Positions of first appearances in A355731.
Counting distinct sequences after sorting: A355734, firsts of A355733.
Requiring the result to be weakly increasing: A355736, firsts of A355735.
Requiring the result to be relatively prime: A355738, firsts of A355737.
A000005 counts divisors.
A001414 adds up distinct prime divisors, counted by A001221.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
Sequence in context: A128052 A033681 A074339 * A115164 A088801 A003033
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 21 2022
STATUS
approved