A355736 - OEIS

A355736

Least k such that there are exactly n ways to choose a divisor of each prime index of k (taken in weakly increasing order) such that the result is also weakly increasing.

1, 3, 7, 13, 21, 37, 39, 89, 133, 117, 111, 273, 351, 259, 267, 333, 453, 793, 669, 623, 999, 777, 843, 1491, 1157, 1561, 2863, 1443, 1963, 2331, 1977, 1869, 2899, 2529, 3207, 4107, 3171, 5073, 4329, 3653, 4667, 3471, 7399, 4613, 7587, 5931, 7269, 5889, 7483

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OFFSET

1,2

COMMENTS

This is the position of first appearance of n in A355735.

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.

LINKS

Table of n, a(n) for n=1..49.

EXAMPLE

The terms together with their prime indices begin:

1: {}

3: {2}

7: {4}

13: {6}

21: {2,4}

37: {12}

39: {2,6}

89: {24}

133: {4,8}

117: {2,2,6}

111: {2,12}

273: {2,4,6}

351: {2,2,2,6}

For example, the choices for a(12) = 273 are:

{1,1,1} {1,2,2} {2,2,2}

{1,1,2} {1,2,3} {2,2,3}

{1,1,3} {1,2,6} {2,2,6}

{1,1,6} {1,4,6} {2,4,6}

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]], LessEqual@@#&]], {n, 1000}];

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

CROSSREFS

Allowing any choice of divisors gives A355732, firsts of A355731.

Choosing a multiset instead of sequence gives A355734, firsts of A355733.

Positions of first appearances in A355735.

The case of prime factors instead of divisors is counted by A355745.

The decreasing version is counted by A355749.

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.

Cf. A000720, A076610, A355737, A355739, A355740, A355741, A355744.

Sequence in context: A098575 A363144 A138035 * A032606 A098482 A342422

Adjacent sequences: A355733 A355734 A355735 * A355737 A355738 A355739

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 21 2022

STATUS

approved