A308299 Numbers whose prime indices are factorial numbers.

1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18, 24, 26, 27, 32, 36, 39, 48, 52, 54, 64, 72, 78, 81, 89, 96, 104, 108, 117, 128, 144, 156, 162, 169, 178, 192, 208, 216, 234, 243, 256, 267, 288, 312, 324, 338, 351, 356, 384, 416, 432, 468, 486, 507, 512, 534, 576, 624, 648

Offset

1,2

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. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions using factorial numbers. The enumeration of these partitions by sum is given by A064986.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(k!)) = 3.292606708493... . - Amiram Eldar, Dec 03 2022

Example

The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
    9: {2,2}
   12: {1,1,2}
   13: {6}
   16: {1,1,1,1}
   18: {1,2,2}
   24: {1,1,1,2}
   26: {1,6}
   27: {2,2,2}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   39: {2,6}
   48: {1,1,1,1,2}
   52: {1,1,6}
   54: {1,2,2,2}

Mathematica

nn=5;
facts=Array[Factorial, nn];
Select[Range[Prime[Max@@facts]], SubsetQ[facts, PrimePi/@First/@FactorInteger[#]]&]

Crossrefs

Cf. A000142, A056239, A062439, A064986, A112798, A115944, A284605, A322583, A325616, A325709, A325618.

Sequence in context: A035240 A278580 A028929 * A356940 A274452 A018731

Adjacent sequences: A308296 A308297 A308298 * A308300 A308301 A308302

Keyword

nonn

Author

Gus Wiseman, May 19 2019

Status

approved