A363951 Numbers whose prime indices satisfy (length) = (mean), or (sum) = (length)^2.

2, 9, 10, 68, 78, 98, 99, 105, 110, 125, 328, 444, 558, 620, 783, 812, 870, 966, 988, 1012, 1035, 1150, 1156, 1168, 1197, 1254, 1326, 1330, 1425, 1521, 1666, 1683, 1690, 1704, 1785, 1870, 1911, 2002, 2125, 2145, 2275, 2401, 2412, 2541, 2662, 2680, 2695, 3025

Offset: 1

Author

Gus Wiseman, Jul 05 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:
    2: {1}
    9: {2,2}
   10: {1,3}
   68: {1,1,7}
   78: {1,2,6}
   98: {1,4,4}
   99: {2,2,5}
  105: {2,3,4}
  110: {1,3,5}
  125: {3,3,3}
  328: {1,1,1,13}
  444: {1,1,2,12}
  558: {1,2,2,11}
  620: {1,1,3,11}
  783: {2,2,2,10}
  812: {1,1,4,10}
  870: {1,2,3,10}
  966: {1,2,4,9}
  988: {1,1,6,8}
		

Crossrefs

Partitions of this type are counted by A364055, without zeros A206240.

The RHS is A001222.

The LHS is A326567/A326568.

A008284 counts partitions by length, A058398 by mean.

A088529/A088530 gives mean of prime signature A124010.

A112798 lists prime indices, sum A056239.

A124943 counts partitions by low median, high A124944.

A316413 ranks partitions with integer mean, counted by A067538.

A326622 counts factorizations with integer mean, strict A328966.

A363950 ranks partitions with low mean 2, counted by A026905 redoubled.

Cf. A025065, A327473, A327476, A327482, A359889, A363723, A363727, A363729, A363944, A363949.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Mean[prix[#]]==PrimeOmega[#]&]