login
A324521
Numbers > 1 where the maximum prime index is less than or equal to the number of prime factors counted with multiplicity.
31
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
1,1
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.
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).
FORMULA
A061395(a(n)) <= A001222(a(n)).
EXAMPLE
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}
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[#]&]
PROG
(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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 06 2019
STATUS
approved