cp's OEIS Frontend

A111907 Numbers k such that the same number of primes, among primes <= the largest prime dividing k, divide k as do not.

%I A111907 #25 Jun 21 2024 07:35:03
%S A111907 1,3,9,14,21,27,28,35,56,63,78,81,98,112,130,147,156,175,182,189,195,
%T A111907 196,224,234,243,245,260,273,286,312,364,392,429,441,448,455,468,520,
%U A111907 567,570,572,585,624,650,686,702,715,728,729,784,798,819,875,896,936
%N A111907 Numbers k such that the same number of primes, among primes <= the largest prime dividing k, divide k as do not.
%C A111907 Also numbers whose greatest prime index (A061395) is twice their number of distinct prime factors (A001221). - _Gus Wiseman_, Mar 19 2023
%H A111907 John Tyler Rascoe, <a href="/A111907/b111907.txt">Table of n, a(n) for n = 1..1000</a>
%e A111907 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 A111907 From _Gus Wiseman_, Mar 19 2023: (Start)
%e A111907 The terms together with their prime indices begin:
%e A111907     1: {}
%e A111907     3: {2}
%e A111907     9: {2,2}
%e A111907    14: {1,4}
%e A111907    21: {2,4}
%e A111907    27: {2,2,2}
%e A111907    28: {1,1,4}
%e A111907    35: {3,4}
%e A111907    56: {1,1,1,4}
%e A111907    63: {2,2,4}
%e A111907    78: {1,2,6}
%e A111907    81: {2,2,2,2}
%e A111907    98: {1,4,4}
%e A111907   112: {1,1,1,1,4}
%e A111907   130: {1,3,6}
%e A111907   147: {2,4,4}
%e A111907   156: {1,1,2,6}
%e A111907 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 A111907 (End)
%t A111907 Select[Range[100],2*PrimeNu[#]==PrimePi[FactorInteger[#][[-1,1]]]&] (* _Gus Wiseman_, Mar 19 2023 *)
%o A111907 (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 A111907 (Python)
%o A111907 from itertools import count, islice
%o A111907 from sympy import sieve, factorint
%o A111907 def a_gen():
%o A111907     yield 1
%o A111907     for k in count(3):
%o A111907         f = [sieve.search(i)[0] for i in factorint(k)]
%o A111907         if 2*len(f) == f[-1]:
%o A111907             yield k
%o A111907 A111907_list = list(islice(a_gen(), 100)) # _John Tyler Rascoe_, Jun 20 2024
%Y A111907 For length instead of maximum we have A067801.
%Y A111907 These partitions are counted by A239959.
%Y A111907 A001222 (bigomega) counts prime factors, distinct A001221 (omega).
%Y A111907 A061395 gives greatest prime index.
%Y A111907 A112798 lists prime indices, sum A056239.
%Y A111907 Comparing twice the number of distinct parts to greatest part:
%Y A111907               less: A360254, ranks A111906
%Y A111907              equal: A239959, ranks A111907
%Y A111907            greater: A237365, ranks A111905
%Y A111907      less or equal: A237363, ranks A361204
%Y A111907   greater or equal: A361394, ranks A361395
%Y A111907 Cf. A046660, A067340, A324517, A324521, A324562, A361205, A361393.
%K A111907 nonn
%O A111907 1,2
%A A111907 _Leroy Quet_, Aug 19 2005
%E A111907 More terms from _Klaus Brockhaus_, Aug 21 2005