A335241 Numbers whose prime indices are not pairwise coprime, where a singleton is not coprime unless it is {1}.

1, 3, 5, 7, 9, 11, 13, 17, 18, 19, 21, 23, 25, 27, 29, 31, 36, 37, 39, 41, 42, 43, 45, 47, 49, 50, 53, 54, 57, 59, 61, 63, 65, 67, 71, 72, 73, 75, 78, 79, 81, 83, 84, 87, 89, 90, 91, 97, 98, 99, 100, 101, 103, 105, 107, 108, 109, 111, 113, 114, 115, 117, 121

Offset: 1

Author

Gus Wiseman, May 30 2020

Keywords

• nonn

Comments

We use the Mathematica definition for CoprimeQ, so a singleton is not considered coprime unless it is (1).

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.

Examples

			The sequence of terms together with their prime indices begins:
    1: {}          31: {11}          61: {18}
    3: {2}         36: {1,1,2,2}     63: {2,2,4}
    5: {3}         37: {12}          65: {3,6}
    7: {4}         39: {2,6}         67: {19}
    9: {2,2}       41: {13}          71: {20}
   11: {5}         42: {1,2,4}       72: {1,1,1,2,2}
   13: {6}         43: {14}          73: {21}
   17: {7}         45: {2,2,3}       75: {2,3,3}
   18: {1,2,2}     47: {15}          78: {1,2,6}
   19: {8}         49: {4,4}         79: {22}
   21: {2,4}       50: {1,3,3}       81: {2,2,2,2}
   23: {9}         53: {16}          83: {23}
   25: {3,3}       54: {1,2,2,2}     84: {1,1,2,4}
   27: {2,2,2}     57: {2,8}         87: {2,10}
   29: {10}        59: {17}          89: {24}
		

Crossrefs

The complement is A302696.

The version for relatively prime instead of coprime is A318978.

The version for standard compositions is A335239.

These are the Heinz numbers of the partitions counted by A335240.

Singleton or pairwise coprime partitions are counted by A051424.

Singleton or pairwise coprime sets are ranked by A087087.

Primes and numbers with pairwise coprime prime indices are A302569.

Numbers whose binary indices are pairwise coprime are A326675.

Coprime standard composition numbers are A333227.

Cf. A007360, A101268, A326674, A327516, A333228, A335236, A335237, A335238.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!CoprimeQ@@primeMS[#]&]