A324521 Numbers > 1 where the maximum prime index is less than or equal to the number of prime factors counted with multiplicity.

2, 4, 6, 8, 9, 12, 16, 18, 20, 24, 27, 30, 32, 36, 40, 45, 48, 50, 54, 56, 60, 64, 72, 75, 80, 81, 84, 90, 96, 100, 108, 112, 120, 125, 126, 128, 135, 140, 144, 150, 160, 162, 168, 176, 180, 189, 192, 196, 200, 210, 216, 224, 225, 240, 243, 250, 252, 256

Offset: 1

Author

Gus Wiseman, Mar 06 2019

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.

Also Heinz numbers of integer partitions with nonnegative rank (A064174). The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Examples

			The sequence of terms together with their prime indices begins:
   2: {1}
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
   9: {2,2}
  12: {1,1,2}
  16: {1,1,1,1}
  18: {1,2,2}
  20: {1,1,3}
  24: {1,1,1,2}
  27: {2,2,2}
  30: {1,2,3}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  40: {1,1,1,3}
  45: {2,2,3}
  48: {1,1,1,1,2}
		

Links

• Matthieu Pluntz, Table of n, a(n) for n = 1..10929 (up to a(n) = 2^21)

Crossrefs

Cf. A001222, A003114, A056239, A061395, A064174, A106529, A112798, A256617.

Cf. A324515, A324517, A324519, A324522, A324560, A324562.

Programs

Maple

with(numtheory):
q:= n-> is(pi(max(factorset(n)))<=bigomega(n)):
select(q, [$2..300])[];  # Alois P. Heinz, Mar 07 2019

Mathematica

Select[Range[2,100],PrimePi[FactorInteger[#][[-1,1]]]<=PrimeOmega[#]&]

PARI

isok(m) = (m>1) && (primepi(vecmax(factor(m)[, 1])) <= bigomega(m)); \\ Michel Marcus, Nov 14 2022

Python

from sympy import factorint, primepi
def ok(n):
    f = factorint(n)
    return primepi(max(f)) <= sum(f.values())
print([k for k in range(2, 257) if ok(k)]) # Michael S. Branicky, Nov 15 2022

Formula

A061395(a(n)) <= A001222(a(n)).