A360008 Positions of first appearances in the sequence giving the mean of prime indices (A326567/A326568).

1, 3, 5, 6, 7, 11, 12, 13, 14, 17, 18, 19, 23, 24, 26, 29, 31, 37, 38, 41, 42, 43, 47, 48, 52, 53, 54, 58, 59, 61, 67, 71, 72, 73, 74, 76, 79, 83, 86, 89, 92, 96, 97, 101, 103, 104, 106, 107, 108, 109, 113, 122, 124, 127, 131, 137, 139, 142, 148, 149, 151, 152

Offset: 1

Author

Gus Wiseman, Jan 24 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.

Examples

			The terms together with their prime indices begin:
    1: {}
    3: {2}
    5: {3}
    6: {1,2}
    7: {4}
   11: {5}
   12: {1,1,2}
   13: {6}
   14: {1,4}
   17: {7}
   18: {1,2,2}
   19: {8}
   23: {9}
   24: {1,1,1,2}
		

Crossrefs

Positions of first appearances in A326567/A326568.

The version for median instead of mean is A360007, unsorted A360006.

A058398 counts partitions by mean, see also A008284, A327482.

A112798 lists prime indices, length A001222, sum A056239.

A316413 lists numbers whose prime indices have integer mean.

A326567/A326568 gives mean of prime indices.

A359908 = numbers w/ integer median of prime indices, complement A359912.

Cf. A026424, A051293, A327473, A348551, A359889, A360005.

Programs

Mathematica

nn=1000;
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
seq=Table[If[n==1,1,Mean[prix[n]]],{n,nn}];
Select[Range[nn],FreeQ[seq[[Range[#-1]]],seq[[#]]]&]