A338330 - OEIS

%I #10 Nov 20 2020 17:16:52

%S 21,39,42,57,63,65,78,84,87,91,105,111,114,115,117,126,129,130,133,

%T 147,156,159,168,171,174,182,183,185,189,195,203,210,213,222,228,230,

%U 231,234,235,237,247,252,258,259,260,261,266,267,273,285,294,299,301

%N Numbers that are neither a power of a prime (A000961) nor is their set of distinct prime indices pairwise coprime.

%C Also Heinz numbers of partitions that are neither constant (A144300) nor have pairwise coprime distinct parts (A304709), hence the formula. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

%F Equals A024619 \ A304711.

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

%e      21: {2,4}        126: {1,2,2,4}      203: {4,10}

%e      39: {2,6}        129: {2,14}         210: {1,2,3,4}

%e      42: {1,2,4}      130: {1,3,6}        213: {2,20}

%e      57: {2,8}        133: {4,8}          222: {1,2,12}

%e      63: {2,2,4}      147: {2,4,4}        228: {1,1,2,8}

%e      65: {3,6}        156: {1,1,2,6}      230: {1,3,9}

%e      78: {1,2,6}      159: {2,16}         231: {2,4,5}

%e      84: {1,1,2,4}    168: {1,1,1,2,4}    234: {1,2,2,6}

%e      87: {2,10}       171: {2,2,8}        235: {3,15}

%e      91: {4,6}        174: {1,2,10}       237: {2,22}

%e     105: {2,3,4}      182: {1,4,6}        247: {6,8}

%e     111: {2,12}       183: {2,18}         252: {1,1,2,2,4}

%e     114: {1,2,8}      185: {3,12}         258: {1,2,14}

%e     115: {3,9}        189: {2,2,2,4}      259: {4,12}

%e     117: {2,2,6}      195: {2,3,6}        260: {1,1,3,6}

%t Select[Range[2,100],!PrimePowerQ[#]&&!CoprimeQ@@Union[PrimePi/@First/@FactorInteger[#]]&]

%Y A338331 is the complement.

%Y A304713 is the complement of the version for divisibility.

%Y Cf. A056239, A112798, A289509, A303282, A316476, A318716, A318719, A328867.

%K nonn

%O 1,1

%A _Gus Wiseman_, Nov 12 2020