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Nonprime numbers whose prime indices have a common divisor > 1.
3

%I #5 Sep 23 2019 13:43:40

%S 9,21,25,27,39,49,57,63,65,81,87,91,111,115,117,121,125,129,133,147,

%T 159,169,171,183,185,189,203,213,235,237,243,247,259,261,267,273,289,

%U 299,301,303,305,319,321,325,333,339,343,351,361,365,371,377,387,393,399

%N Nonprime numbers whose prime indices have a common divisor > 1.

%C First differs from A322336 in lacking 2535 = prime(2)*prime(3)*prime(6)^2.

%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. Heinz numbers of integer partitions with a common divisor > 1 are A318978, and the enumeration of these partitions by sum is A108572.

%F Complement of A000040 in A318978.

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

%e 9: {2,2}

%e 21: {2,4}

%e 25: {3,3}

%e 27: {2,2,2}

%e 39: {2,6}

%e 49: {4,4}

%e 57: {2,8}

%e 63: {2,2,4}

%e 65: {3,6}

%e 81: {2,2,2,2}

%e 87: {2,10}

%e 91: {4,6}

%e 111: {2,12}

%e 115: {3,9}

%e 117: {2,2,6}

%e 121: {5,5}

%e 125: {3,3,3}

%e 129: {2,14}

%e 133: {4,8}

%e 147: {2,4,4}

%t Select[Range[100],#>1&&!PrimeQ[#]&&GCD@@PrimePi/@First/@FactorInteger[#]>1&]

%Y Cf. A000837, A007916, A018783, A056239, A108572, A112798, A289509, A318978, A327407, A327658.

%K nonn

%O 1,1

%A _Gus Wiseman_, Sep 22 2019