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

1,1

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.

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)).

Example

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.

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[#]]&]

Prog

(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

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.

Sequence in context: A031195 A034117 A104863 * A326122 A027183 A333300

Adjacent sequences: A362044 A362045 A362046 * A362048 A362049 A362050

Keyword

nonn

Author

Gus Wiseman, Apr 11 2023

Status

approved