A275199 Numbers having equal numbers of distinct prime factors of forms 6*k+1 and 6*k+5.

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 35, 36, 48, 54, 64, 65, 70, 72, 77, 81, 95, 96, 105, 108, 119, 128, 130, 140, 143, 144, 154, 155, 161, 162, 175, 185, 190, 192, 195, 203, 209, 210, 215, 216, 221, 231, 238, 243, 245, 256, 260, 280, 285, 286, 287

Offset

1,2

Comments

This sequence and A275200 and A275201 partition the positive integers.

Links

Clark Kimberling, Table of n, a(n) for n = 1..1000

Example

35 = 5^1 7^1, in which the number of distinct primes 6*k+1 is 1 and the number of distinct primes 6*k + 5 is 1.

Maple

N:= 1000: # to get all terms <= N
filter:= proc(n)
  local P1, P5;
P1, P5:= selectremove(t -> t mod 6 = 1, numtheory:-factorset(n) minus {2, 3});
nops(P1)=nops(P5)
end proc:
sort(map(t -> seq(t*2^j, j=0..ilog2(N/t)),
select(filter, [seq(i, i=1..N, 2)]))); # Robert Israel, Jul 20 2016

Mathematica

g[n_] := Map[First, FactorInteger[n]];
p1 = Select[Prime[Range[200]], Mod[#, 6] == 1 &];
p2 = Select[Prime[Range[200]], Mod[#, 6] == 5 &];
q1[n_] := Length[Intersection[g[n], p1]]
q2[n_] := Length[Intersection[g[n], p2]]
Select[Range[200], q1[#] == q2[#] &] (* A275199 *)
Select[Range[200], q1[#] < q2[#] &]  (* A275200 *)
Select[Range[200], q1[#] > q2[#] &]  (* A275201 *)

Crossrefs

Cf. A275200, A275201.

Sequence in context: A097755 A301704 A083854 * A003586 A114334 A262609

Adjacent sequences: A275196 A275197 A275198 * A275200 A275201 A275202

Keyword

nonn,easy

Author

Clark Kimberling, Jul 20 2016

Status

approved