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

1,2

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.

Links

Table of n, a(n) for n=1..63.

Example

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}

Mathematica

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

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.

Sequence in context: A247428 A184591 A033040 * A200982 A084820 A324846

Adjacent sequences: A335238 A335239 A335240 * A335242 A335243 A335244

Keyword

nonn

Author

Gus Wiseman, May 30 2020

Status

approved