A338330 Numbers that are neither a power of a prime (A000961) nor is their set of distinct prime indices pairwise coprime.

21, 39, 42, 57, 63, 65, 78, 84, 87, 91, 105, 111, 114, 115, 117, 126, 129, 130, 133, 147, 156, 159, 168, 171, 174, 182, 183, 185, 189, 195, 203, 210, 213, 222, 228, 230, 231, 234, 235, 237, 247, 252, 258, 259, 260, 261, 266, 267, 273, 285, 294, 299, 301

Offset: 1

Author

Gus Wiseman, Nov 12 2020

Keywords

• nonn

Comments

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.

Examples

			The sequence of terms together with their prime indices begins:
     21: {2,4}        126: {1,2,2,4}      203: {4,10}
     39: {2,6}        129: {2,14}         210: {1,2,3,4}
     42: {1,2,4}      130: {1,3,6}        213: {2,20}
     57: {2,8}        133: {4,8}          222: {1,2,12}
     63: {2,2,4}      147: {2,4,4}        228: {1,1,2,8}
     65: {3,6}        156: {1,1,2,6}      230: {1,3,9}
     78: {1,2,6}      159: {2,16}         231: {2,4,5}
     84: {1,1,2,4}    168: {1,1,1,2,4}    234: {1,2,2,6}
     87: {2,10}       171: {2,2,8}        235: {3,15}
     91: {4,6}        174: {1,2,10}       237: {2,22}
    105: {2,3,4}      182: {1,4,6}        247: {6,8}
    111: {2,12}       183: {2,18}         252: {1,1,2,2,4}
    114: {1,2,8}      185: {3,12}         258: {1,2,14}
    115: {3,9}        189: {2,2,2,4}      259: {4,12}
    117: {2,2,6}      195: {2,3,6}        260: {1,1,3,6}
		

Crossrefs

A338331 is the complement.

A304713 is the complement of the version for divisibility.

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

Programs

Mathematica

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

Formula

Equals A024619 \ A304711.