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

1,1

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.

Links

Table of n, a(n) for n=1..48.

Example

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}

Mathematica

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

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.

Sequence in context: A191401 A363223 A338997 * A085069 A371163 A207670

Adjacent sequences: A363948 A363949 A363950 * A363952 A363953 A363954

Keyword

nonn

Author

Gus Wiseman, Jul 05 2023

Status

approved