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

Author

Gus Wiseman, May 19 2019

Keywords

• nonn

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.

Examples

			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}
		

Links

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

Crossrefs

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

Programs

Mathematica

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

Formula

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