A371131 Least number with exactly n distinct divisors of prime indices. Position of first appearance of n in A370820.

1, 2, 3, 7, 13, 53, 37, 311, 89, 151, 223, 2045, 281, 3241, 1163, 827, 659, 9037, 1069, 17611, 1511, 4211, 28181, 122119, 2423, 10627, 88483, 6997, 7561, 98965, 5443, 88099, 6473, 95603, 309073, 50543, 10271, 192709, 508051, 438979, 14323, 305107, 26203

Offset

0,2

Comments

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.

Every nonnegative integer belongs to A370820, so this sequence is infinite.

Are there any terms with more than two prime factors?

Links

Table of n, a(n) for n=0..42.

Example

The terms together with their prime indices begin:
       1: {}
       2: {1}
       3: {2}
       7: {4}
      13: {6}
      53: {16}
      37: {12}
     311: {64}
      89: {24}
     151: {36}
     223: {48}
    2045: {3,80}
     281: {60}
    3241: {4,90}
    1163: {192}
     827: {144}
     659: {120}
    9037: {4,210}
    1069: {180}
   17611: {5,252}

Mathematica

rnnm[q_]:=Max@@Select[Range[Min@@q, Max@@q], SubsetQ[q, Range[#]]&];
posfirsts[q_]:=Table[Position[q, n][[1, 1]], {n, Min@@q, rnnm[q]}];
posfirsts[Table[Length[Union @@ Divisors/@PrimePi/@First/@If[n==1, {}, FactorInteger[n]]], {n, 1000}]]

Prog

(PARI) f(n) = my(list=List(), f=factor(n)); for (i=1, #f~, fordiv(primepi(f[i, 1]), d, listput(list, d))); #Set(list); \\ A370820
a(n) = my(k=1); while (f(k) != n, k++); k; \\ Michel Marcus, May 02 2024

Crossrefs

Counting prime factors instead of divisors (see A303975) gives A062447(>0).

The sorted version is A371181.

A000005 counts divisors.

A001221 counts distinct prime factors.

A003963 gives product of prime indices.

A027746 lists prime factors, A112798 indices, length A001222.

A355731 counts choices of a divisor of each prime index, firsts A355732.

A355741 counts choices of a prime factor of each prime index.

Cf. A000720, A000792, A005179, A007416, A355739, A370348, A370802, A370808, A371130, A371165, A371177.

Sequence in context: A166945 A257393 A273814 * A361988 A085872 A070858

Adjacent sequences: A371128 A371129 A371130 * A371132 A371133 A371134

Keyword

nonn

Author

Gus Wiseman, Mar 20 2024

Status

approved