A324521 - OEIS

%I #27 Dec 21 2022 22:05:38

%S 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 80,81,84,90,96,100,108,112,120,125,126,128,135,140,144,150,160,162,

%U 168,176,180,189,192,196,200,210,216,224,225,240,243,250,252,256

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

%C 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 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 Matthieu Pluntz, <a href="/A324521/b324521.txt">Table of n, a(n) for n = 1..10929 (up to a(n) = 2^21)</a>

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

%e The sequence of terms together with their prime indices begins:

%e    2: {1}

%e    4: {1,1}

%e    6: {1,2}

%e    8: {1,1,1}

%e    9: {2,2}

%e   12: {1,1,2}

%e   16: {1,1,1,1}

%e   18: {1,2,2}

%e   20: {1,1,3}

%e   24: {1,1,1,2}

%e   27: {2,2,2}

%e   30: {1,2,3}

%e   32: {1,1,1,1,1}

%e   36: {1,1,2,2}

%e   40: {1,1,1,3}

%e   45: {2,2,3}

%e   48: {1,1,1,1,2}

%p with(numtheory):

%p q:= n-> is(pi(max(factorset(n)))<=bigomega(n)):

%p select(q, [$2..300])[];  # _Alois P. Heinz_, Mar 07 2019

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

%o (PARI) isok(m) = (m>1) && (primepi(vecmax(factor(m)[, 1])) <= bigomega(m)); \\ _Michel Marcus_, Nov 14 2022

%o (Python)

%o from sympy import factorint, primepi

%o def ok(n):

%o     f = factorint(n)

%o     return primepi(max(f)) <= sum(f.values())

%o print([k for k in range(2, 257) if ok(k)]) # _Michael S. Branicky_, Nov 15 2022

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

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

%K nonn

%O 1,1

%A _Gus Wiseman_, Mar 06 2019