A360669 Nonprime numbers > 1 for which the prime indices have the same mean as their first differences.

10, 39, 68, 115, 138, 259, 310, 328, 387, 517, 574, 636, 793, 795, 1034, 1168, 1206, 1241, 1281, 1340, 1534, 1691, 1825, 2212, 2278, 2328, 2343, 2369, 2370, 2727, 2774, 2905, 3081, 3277, 3818, 3924, 4064, 4074, 4247, 4268, 4360, 4539, 4850, 4905, 5243, 5335

Offset: 1

Author

Gus Wiseman, Feb 18 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: {}
    10: {1,3}
    39: {2,6}
    68: {1,1,7}
   115: {3,9}
   138: {1,2,9}
   259: {4,12}
   310: {1,3,11}
   328: {1,1,1,13}
   387: {2,2,14}
   517: {5,15}
   574: {1,4,13}
   636: {1,1,2,16}
For example, the prime indices of 138 are {1,2,9}, with mean 4, and with first differences (1,7), with mean also 4, so 138 is in the sequence.
		

Crossrefs

These partitions are counted by A360670.

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

A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.

A124010 gives prime signature, mean A088529/A088530.

A301987 lists numbers whose sum of prime indices equals their product.

A316413 lists numbers whose prime indices have integer mean.

A334201 adds up all prime indices except the greatest.

A360614/A360615 = mean of first differences of 0-prepended prime indices.

Cf. A340610, A344415, A348551, A359904, A360008, A360068, A360681.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,1000],Mean[prix[#]]==Mean[Differences[prix[#]]]&]