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A286679 Numbers of the form (2*prime(n)^2 + 1)/3. 1
17, 33, 81, 113, 193, 241, 353, 561, 641, 913, 1121, 1233, 1473, 1873, 2321, 2481, 2993, 3361, 3553, 4161, 4593, 5281, 6273, 6801, 7073, 7633, 7921, 8513, 10753, 11441, 12513, 12881, 14801, 15201, 16433, 17713, 18593, 19953, 21361, 21841 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,1

COMMENTS

Aside from p = 3, p = 1 or 2 mod 3 and p^2 = 1 mod 3. Thus 2p^2 + 1 is a multiple of 3.

LINKS

G. C. Greubel, Table of n, a(n) for n = 3..1000

FORMULA

Product_{n >= 3} (3a(n) + 1) / (3a(n) - 1) = (26/25) * (50/49) * (122/121) * ... = 54/(5Pi^2) = 1.0942687833372479315938982026650585002 (constant).

a(3) = 17; a(n + 1) = a(n) + 16 * A075888(n-2) for n > 3.

Numbers of the form  16k + 1 for some k. In particular k belongs to A001318, excluding those who sqrt(24 * A001318(k) + 1) are composites.

MATHEMATICA

(2Prime[Range[3, 50]]^2 + 1)/3 (* Alonso del Arte, May 12 2017 *)

PROG

(PARI) {

forprime(n=5, 300,

         print1((2*n^2+1)/3", ")

        )

}

(MAGMA) [(2*NthPrime(n)^2+1)/3: n in [3..50]]; // Vincenzo Librandi, May 15 2017 *)

CROSSREFS

Cf. A075888, A001318.

Sequence in context: A044443 A158057 A249356 * A116523 A168579 A135637

Adjacent sequences:  A286676 A286677 A286678 * A286680 A286681 A286682

KEYWORD

nonn

AUTHOR

Dimitris Valianatos, May 12 2017

STATUS

approved

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Last modified July 22 10:41 EDT 2019. Contains 325219 sequences. (Running on oeis4.)