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%I #14 Mar 20 2023 21:07:24
%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 A number n is included if the same number of primes, among primes <= the largest prime dividing n, divide n 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
%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)
%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
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