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A173178
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Numbers k such that 2*k+3 is a prime of the form 3*A024893(m) + 2.
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2
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1, 4, 7, 10, 13, 19, 22, 25, 28, 34, 40, 43, 49, 52, 55, 64, 67, 73, 82, 85, 88, 94, 97, 112, 115, 118, 124, 127, 130, 133, 139, 145, 154, 157, 172, 175, 178, 190, 193, 199, 208, 214, 220, 223, 229, 232, 238, 244, 250, 253, 259, 277, 280, 283, 292, 295, 298, 307, 319
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OFFSET
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1,2
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COMMENTS
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With the Bachet-Bézout theorem implicating Gauss Lemma and the Fundamental Theorem of Arithmetic,
for k > 1, k = 2*a + 3*b (a and b integers)
first type
A024893 Numbers k such that 3*k + 2 is prime
A034936 Numbers k such that 3*k + 4 is prime
OR second type
A067076 Numbers k such that 2*k + 3 is prime
k a b OR a b
-- - - - -
0 0 0 0 0
1 - - - -
2 1 0 1 0
3 0 1 0 1
4 2 0 2 0
5 1 1 1 1
6 0 2 3 0
7 2 1 2 1
8 1 2 4 0
9 0 3 3 1
10 2 2 5 0
11 1 3 4 1
12 0 4 6 0
13 2 3 5 1
14 1 4 7 0
15 0 5 6 1
...
2* 1 + 3 OR 3* 1 + 2 = 5;
2* 4 + 3 OR 3* 3 + 2 = 11;
2* 7 + 3 OR 3* 5 + 2 = 17;
2*10 + 3 OR 3* 7 + 2 = 23;
2*13 + 3 OR 3* 9 + 2 = 29;
2*19 + 3 OR 3*13 + 2 = 41;
2*22 + 3 OR 3*15 + 2 = 47;
2*25 + 3 OR 3*17 + 2 = 53;
2*28 + 3 OR 3*19 + 2 = 59.
A024893 Numbers k such that 3k+2 is prime.
A024898 Positive integers k such that 6k-1 is prime.
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LINKS
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FORMULA
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MATHEMATICA
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Select[Range[0, 320], PrimeQ[(p = 2*# + 3)] && Mod[p, 3] == 2 &] (* Amiram Eldar, Jul 30 2024 *)
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CROSSREFS
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KEYWORD
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nonn,easy,uned
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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