A366847 Numbers whose halved even prime indices are nonempty and relatively prime.

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 91, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162, 165, 168, 171, 174

Offset: 1

Author

Gus Wiseman, Oct 31 2023

Keywords

• nonn

Comments

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.

Consists of powers of 2 times elements of the odd restriction A366849.

Examples

			The even prime indices of 91 are {4,6}, halved {2,3}, which are relatively prime, so 91 is in the sequence.
The prime indices of 665 are {3,4,8}, even {4,8}, halved {2,4}, which are not relatively prime, so 665 is not in the sequence.
The terms together with their prime indices begin:
    3: {2}
    6: {1,2}
    9: {2,2}
   12: {1,1,2}
   15: {2,3}
   18: {1,2,2}
   21: {2,4}
   24: {1,1,1,2}
   27: {2,2,2}
   30: {1,2,3}
   33: {2,5}
   36: {1,1,2,2}
   39: {2,6}
   42: {1,2,4}
   45: {2,2,3}
   48: {1,1,1,1,2}
		

Crossrefs

Including odd indices gives A289509, ones of A289508, counted by A000837.

The complement including odd indices is A318978, counted by A018783.

The partitions with these ranks are counted by A366845.

A version for odd indices A366846, counted by A366850.

The odd restriction is A366849.

A000041 counts integer partitions, strict A000009 (also into odds).

A035363 counts partitions into all even parts, ranks A066207.

A112798 lists prime indices, length A001222, sum A056239.

A162641 counts even prime exponents, odd A162642.

A257992 counts even prime indices, odd A257991.

A366528 adds up odd prime indices, partition triangle A113685.

A366531 = 2*A366533 adds up even prime indices, triangle A113686/A174713.

Cf. A000720, A055396, A061395, A066208, A168532, A302696, A302697, A325698, A366842, A366843, A366844, A366848.

Programs

Mathematica

Select[Range[100],GCD@@Select[PrimePi/@First/@FactorInteger[#],EvenQ]/2==1&]