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Numbers whose prime indices are factorial numbers.
2

%I #13 Dec 03 2022 05:45:31

%S 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 96,104,108,117,128,144,156,162,169,178,192,208,216,234,243,256,267,

%U 288,312,324,338,351,356,384,416,432,468,486,507,512,534,576,624,648

%N Numbers whose prime indices are factorial numbers.

%C 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 Amiram Eldar, <a href="/A308299/b308299.txt">Table of n, a(n) for n = 1..10000</a>

%F Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(k!)) = 3.292606708493... . - _Amiram Eldar_, Dec 03 2022

%e The sequence of terms together with their prime indices begins:

%e 1: {}

%e 2: {1}

%e 3: {2}

%e 4: {1,1}

%e 6: {1,2}

%e 8: {1,1,1}

%e 9: {2,2}

%e 12: {1,1,2}

%e 13: {6}

%e 16: {1,1,1,1}

%e 18: {1,2,2}

%e 24: {1,1,1,2}

%e 26: {1,6}

%e 27: {2,2,2}

%e 32: {1,1,1,1,1}

%e 36: {1,1,2,2}

%e 39: {2,6}

%e 48: {1,1,1,1,2}

%e 52: {1,1,6}

%e 54: {1,2,2,2}

%t nn=5;

%t facts=Array[Factorial,nn];

%t Select[Range[Prime[Max@@facts]],SubsetQ[facts,PrimePi/@First/@FactorInteger[#]]&]

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

%K nonn

%O 1,2

%A _Gus Wiseman_, May 19 2019