A111907 - OEIS

%I #25 Jun 21 2024 07:35:03

%S 1,3,9,14,21,27,28,35,56,63,78,81,98,112,130,147,156,175,182,189,195,

%T 196,224,234,243,245,260,273,286,312,364,392,429,441,448,455,468,520,

%U 567,570,572,585,624,650,686,702,715,728,729,784,798,819,875,896,936

%N Numbers k such that the same number of primes, among primes <= the largest prime dividing k, divide k as do not.

%C Also numbers whose greatest prime index (A061395) is twice their number of distinct prime factors (A001221). - _Gus Wiseman_, Mar 19 2023

%H John Tyler Rascoe, <a href="/A111907/b111907.txt">Table of n, a(n) for n = 1..1000</a>

%e 28 is included because 7 is the largest prime dividing 28. And of the primes <= 7 (2,3,5,7), 2 and 7 (2 primes) divide 28 and 3 and 5 (also 2 primes) do not divide 28.

%e From _Gus Wiseman_, Mar 19 2023: (Start)

%e The terms together with their prime indices begin:

%e     1: {}

%e     3: {2}

%e     9: {2,2}

%e    14: {1,4}

%e    21: {2,4}

%e    27: {2,2,2}

%e    28: {1,1,4}

%e    35: {3,4}

%e    56: {1,1,1,4}

%e    63: {2,2,4}

%e    78: {1,2,6}

%e    81: {2,2,2,2}

%e    98: {1,4,4}

%e   112: {1,1,1,1,4}

%e   130: {1,3,6}

%e   147: {2,4,4}

%e   156: {1,1,2,6}

%e For example, 156 is included because it has prime indices {1,1,2,6}, with distinct parts {1,2,6} and distinct non-parts {3,4,5}, both of length 3. Alternatively, 156 has greatest prime index 6 and omega 3, and 6 = 2*3.

%e (End)

%t Select[Range[100],2*PrimeNu[#]==PrimePi[FactorInteger[#][[-1,1]]]&] (* _Gus Wiseman_, Mar 19 2023 *)

%o (PARI) {m=950;v=vector(m);for(n=1,m,f=factor(n)[,1]~;c=0;pc=0;forprime(p=2,vecmax(f), j=1;s=length(f);while(j<=s&&p!=f[j],j++);if(j<=s,c++);pc++);v[n]=sign(pc-2*c)); for(n=1,m,if(v[n]==0,print1(n,",")))} \\ _Klaus Brockhaus_, Aug 21 2005

%o (Python)

%o from itertools import count, islice

%o from sympy import sieve, factorint

%o def a_gen():

%o     yield 1

%o     for k in count(3):

%o         f = [sieve.search(i)[0] for i in factorint(k)]

%o         if 2*len(f) == f[-1]:

%o             yield k

%o A111907_list = list(islice(a_gen(), 100)) # _John Tyler Rascoe_, Jun 20 2024

%Y For length instead of maximum we have A067801.

%Y These partitions are counted by A239959.

%Y A001222 (bigomega) counts prime factors, distinct A001221 (omega).

%Y A061395 gives greatest prime index.

%Y A112798 lists prime indices, sum A056239.

%Y Comparing twice the number of distinct parts to greatest part:

%Y               less: A360254, ranks A111906

%Y              equal: A239959, ranks A111907

%Y            greater: A237365, ranks A111905

%Y      less or equal: A237363, ranks A361204

%Y   greater or equal: A361394, ranks A361395

%Y Cf. A046660, A067340, A324517, A324521, A324562, A361205, A361393.

%K nonn

%O 1,2

%A _Leroy Quet_, Aug 19 2005

%E More terms from _Klaus Brockhaus_, Aug 21 2005