cp's OEIS Frontend

A308299 Numbers whose prime indices are factorial numbers.

%I A308299 #13 Dec 03 2022 05:45:31
%S A308299 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,
%T A308299 96,104,108,117,128,144,156,162,169,178,192,208,216,234,243,256,267,
%U A308299 288,312,324,338,351,356,384,416,432,468,486,507,512,534,576,624,648
%N A308299 Numbers whose prime indices are factorial numbers.
%C A308299 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.
%H A308299 Amiram Eldar, <a href="/A308299/b308299.txt">Table of n, a(n) for n = 1..10000</a>
%F A308299 Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(k!)) = 3.292606708493... . - _Amiram Eldar_, Dec 03 2022
%e A308299 The sequence of terms together with their prime indices begins:
%e A308299     1: {}
%e A308299     2: {1}
%e A308299     3: {2}
%e A308299     4: {1,1}
%e A308299     6: {1,2}
%e A308299     8: {1,1,1}
%e A308299     9: {2,2}
%e A308299    12: {1,1,2}
%e A308299    13: {6}
%e A308299    16: {1,1,1,1}
%e A308299    18: {1,2,2}
%e A308299    24: {1,1,1,2}
%e A308299    26: {1,6}
%e A308299    27: {2,2,2}
%e A308299    32: {1,1,1,1,1}
%e A308299    36: {1,1,2,2}
%e A308299    39: {2,6}
%e A308299    48: {1,1,1,1,2}
%e A308299    52: {1,1,6}
%e A308299    54: {1,2,2,2}
%t A308299 nn=5;
%t A308299 facts=Array[Factorial,nn];
%t A308299 Select[Range[Prime[Max@@facts]],SubsetQ[facts,PrimePi/@First/@FactorInteger[#]]&]
%Y A308299 Cf. A000142, A056239, A062439, A064986, A112798, A115944, A284605, A322583, A325616, A325709, A325618.
%K A308299 nonn
%O A308299 1,2
%A A308299 _Gus Wiseman_, May 19 2019