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

1,1

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.

Links

Table of n, a(n) for n=1..53.

Formula

Equals A024619 \ A304711.

Example

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}

Mathematica

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

Crossrefs

A338331 is the complement.

A304713 is the complement of the version for divisibility.

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

Sequence in context: A307278 A176071 A072708 * A102478 A221048 A182297

Adjacent sequences: A338327 A338328 A338329 * A338331 A338332 A338333

Keyword

nonn

Author

Gus Wiseman, Nov 12 2020

Status

approved