cp's OEIS Frontend

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

%I A324521 #27 Dec 21 2022 22:05:38
%S A324521 2,4,6,8,9,12,16,18,20,24,27,30,32,36,40,45,48,50,54,56,60,64,72,75,
%T A324521 80,81,84,90,96,100,108,112,120,125,126,128,135,140,144,150,160,162,
%U A324521 168,176,180,189,192,196,200,210,216,224,225,240,243,250,252,256
%N A324521 Numbers > 1 where the maximum prime index is less than or equal to the number of prime factors counted with multiplicity.
%C A324521 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.
%C A324521 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).
%H A324521 Matthieu Pluntz, <a href="/A324521/b324521.txt">Table of n, a(n) for n = 1..10929 (up to a(n) = 2^21)</a>
%F A324521 A061395(a(n)) <= A001222(a(n)).
%e A324521 The sequence of terms together with their prime indices begins:
%e A324521    2: {1}
%e A324521    4: {1,1}
%e A324521    6: {1,2}
%e A324521    8: {1,1,1}
%e A324521    9: {2,2}
%e A324521   12: {1,1,2}
%e A324521   16: {1,1,1,1}
%e A324521   18: {1,2,2}
%e A324521   20: {1,1,3}
%e A324521   24: {1,1,1,2}
%e A324521   27: {2,2,2}
%e A324521   30: {1,2,3}
%e A324521   32: {1,1,1,1,1}
%e A324521   36: {1,1,2,2}
%e A324521   40: {1,1,1,3}
%e A324521   45: {2,2,3}
%e A324521   48: {1,1,1,1,2}
%p A324521 with(numtheory):
%p A324521 q:= n-> is(pi(max(factorset(n)))<=bigomega(n)):
%p A324521 select(q, [$2..300])[];  # _Alois P. Heinz_, Mar 07 2019
%t A324521 Select[Range[2,100],PrimePi[FactorInteger[#][[-1,1]]]<=PrimeOmega[#]&]
%o A324521 (PARI) isok(m) = (m>1) && (primepi(vecmax(factor(m)[, 1])) <= bigomega(m)); \\ _Michel Marcus_, Nov 14 2022
%o A324521 (Python)
%o A324521 from sympy import factorint, primepi
%o A324521 def ok(n):
%o A324521     f = factorint(n)
%o A324521     return primepi(max(f)) <= sum(f.values())
%o A324521 print([k for k in range(2, 257) if ok(k)]) # _Michael S. Branicky_, Nov 15 2022
%Y A324521 Cf. A001222, A003114, A056239, A061395, A064174, A106529, A112798, A256617.
%Y A324521 Cf. A324515, A324517, A324519, A324522, A324560, A324562.
%K A324521 nonn
%O A324521 1,1
%A A324521 _Gus Wiseman_, Mar 06 2019