A362047 Numbers whose prime indices satisfy: (maximum) - (minimum) = (mean).

10, 30, 39, 90, 98, 99, 100, 115, 259, 270, 273, 300, 490, 495, 517, 663, 665, 793, 810, 900, 1000, 1083, 1241, 1421, 1495, 1521, 1691, 1911, 2058, 2079, 2125, 2145, 2369, 2430, 2450, 2475, 2662, 2700, 2755, 2821, 3000, 3277, 4247, 4495, 4921, 5587, 5863, 6069

Offset: 1

Author

Gus Wiseman, Apr 11 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:
      10: {1,3}
      30: {1,2,3}
      39: {2,6}
      90: {1,2,2,3}
      98: {1,4,4}
      99: {2,2,5}
     100: {1,1,3,3}
     115: {3,9}
     259: {4,12}
     270: {1,2,2,2,3}
     273: {2,4,6}
     300: {1,1,2,3,3}
The prime indices of 490 are {1,3,4,4}, with minimum 1, maximum 4, and mean 3, and 4-1 = 3, so 490 is in the sequence.
		

Crossrefs

Partitions of this type are counted by A361862.

For minimum instead of mean we have A361908, counted by A118096.

A055396 gives minimum prime index, A061395 maximum.

A112798 list prime indices, length A001222, sum A056239.

A243055 subtracts the least prime index from the greatest.

A326844 gives the diagram complement size of Heinz partition.

Cf. A111907, A237832, A268192, A316413, A326836, A326837, A326846, A359358, A359360, A361855.

Programs

Mathematica

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

Python

from itertools import count, islice
from sympy import primepi, factorint
def A362047_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:(primepi(max(f:=factorint(n)))-primepi(min(f)))*sum(f.values())==sum(primepi(i)*j for i, j in f.items()),count(max(startvalue,2)))
A362047_list = list(islice(A362047_gen(),20)) # Chai Wah Wu, Apr 13 2023

Formula

A359360(a(n)) = A326844(a(n)).

A243055(a(n)) = A061395(a(n)) - A055396(a(n))

= A326567(a(n))/A326568(a(n))

= A056239(a(n))/A001222(a(n)).