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Numbers having equal numbers of distinct prime factors of forms 3*k+1 and 3*k+2.
3

%I #17 Jun 09 2022 21:49:36

%S 1,3,9,14,26,27,28,35,38,42,52,56,62,65,74,76,77,78,81,84,86,95,98,

%T 104,105,112,114,119,122,124,126,134,143,146,148,152,155,156,158,161,

%U 168,172,175,185,186,194,195,196,203,206,208,209,215,218,221,222,224

%N Numbers having equal numbers of distinct prime factors of forms 3*k+1 and 3*k+2.

%C This sequence and A274436 and A274437 partition the positive integers.

%H Clark Kimberling, <a href="/A274435/b274435.txt">Table of n, a(n) for n = 1..10000</a>

%e 76 = 2^2 * 19^1, so that the number of distinct primes of the form 3*k+1 is 1 and the number of distinct primes of the form 3*k+2 is 1.

%e 3 and 9 are also terms, since they have no prime factors of either type.

%t g[n_] := Map[First, FactorInteger[n]] ; z = 5000;

%t p1 = Select[Prime[Range[z]], Mod[#, 3] == 1 &];

%t p2 = Select[Prime[Range[z]], Mod[#, 3] == 2 &];

%t q1[n_] := Length[Intersection[g[n], p1]]

%t q2[n_] := Length[Intersection[g[n], p2]]

%t Select[Range[z], q1[#] == q2[#] &];(* A274435 *)

%t Select[Range[z], q1[#] < q2[#] &]; (* A274436 *)

%t Select[Range[z], q1[#] > q2[#] &]; (* A274437 *)

%o (PARI) is(n) = my(f=factor(n)[, 1]~, i=0, j=0); for(k=1, #f, if(!((f[k]-1)%3), i++); if(!((f[k]-2)%3), j++)); i==j \\ _Felix Fröhlich_, Jul 09 2018

%Y Cf. A274436, A274437.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Jul 19 2016