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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.
4
1, 2, 6, 9, 15, 49, 35, 27, 45, 98, 63, 105, 171, 117, 81, 135, 245, 343, 273, 549, 189, 1083, 315, 5618, 741, 686, 507, 513, 351, 243, 405, 7467, 6419, 5575, 735, 6859, 1813, 3231, 1183, 1197, 3537, 819, 1647, 567, 945, 2197, 8397, 3211, 1715, 3249, 3367
OFFSET
1,2
COMMENTS
This is the position of first appearance of n in A355737.
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: {}
2: {1}
6: {1,2}
9: {2,2}
15: {2,3}
49: {4,4}
35: {3,4}
27: {2,2,2}
45: {2,2,3}
98: {1,4,4}
63: {2,2,4}
105: {2,3,4}
171: {2,2,8}
117: {2,2,6}
81: {2,2,2,2}
135: {2,2,2,3}
For example, the choices for a(12) = 105 are:
(1,1,1) (1,3,2) (2,1,4)
(1,1,2) (1,3,4) (2,3,1)
(1,1,4) (2,1,1) (2,3,2)
(1,3,1) (2,1,2) (2,3,4)
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[Length[Select[Tuples[Divisors/@primeMS[n]], GCD@@#==1&]], {n, 100}];
Table[Position[az+1, k][[1, 1]], {k, mnrm[az+1]}]
CROSSREFS
Not requiring coprimality gives A355732, firsts of A355731.
Positions of first appearances in A355737.
A000005 counts divisors.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A120383 lists numbers divisible by all of their prime indices.
A289508 gives GCD of prime indices.
A289509 ranks relatively prime partitions, odd A302697, squarefree A302796.
A324850 lists numbers divisible by the product of their prime indices.
Sequence in context: A181025 A345051 A265202 * A329743 A320496 A172433
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 21 2022
STATUS
approved