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A274435
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Numbers having equal numbers of distinct prime factors of forms 3*k+1 and 3*k+2.
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3
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1, 3, 9, 14, 26, 27, 28, 35, 38, 42, 52, 56, 62, 65, 74, 76, 77, 78, 81, 84, 86, 95, 98, 104, 105, 112, 114, 119, 122, 124, 126, 134, 143, 146, 148, 152, 155, 156, 158, 161, 168, 172, 175, 185, 186, 194, 195, 196, 203, 206, 208, 209, 215, 218, 221, 222, 224
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OFFSET
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1,2
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COMMENTS
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This sequence and A274436 and A274437 partition the positive integers.
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LINKS
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EXAMPLE
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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.
3 and 9 are also terms, since they have no prime factors of either type.
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MATHEMATICA
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g[n_] := Map[First, FactorInteger[n]] ; z = 5000;
p1 = Select[Prime[Range[z]], Mod[#, 3] == 1 &];
p2 = Select[Prime[Range[z]], Mod[#, 3] == 2 &];
q1[n_] := Length[Intersection[g[n], p1]]
q2[n_] := Length[Intersection[g[n], p2]]
Select[Range[z], q1[#] == q2[#] &]; (* A274435 *)
Select[Range[z], q1[#] < q2[#] &]; (* A274436 *)
Select[Range[z], q1[#] > q2[#] &]; (* A274437 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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