A292847 a(n) is the smallest odd prime of the form ((1 + sqrt(2*n))^k - (1 - sqrt(2*n))^k)/(2*sqrt(2*n)).

5, 7, 101, 11, 13, 269, 17, 19, 509, 23, 709, 821, 29, 31, 46957, 55399, 37, 168846239, 41, 43, 9177868096974864412935432937651459122761, 47, 485329129, 2789, 53, 3229, 3461, 59, 61, 1563353111, 139237612541, 67, 5021, 71, 73, 484639, 6221, 79, 6869, 83, 7549

Offset

1,1

Comments

a(146) > 10^1000. - Jason Yuen, Nov 21 2025

Links

Jason Yuen, Table of n, a(n) for n = 1..145

Formula

When 2*n + 3 = p is prime, a(n) = p.

Example

For k = {1, 2, 3, 4, 5}, ((1 + sqrt(6))^k - (1 - sqrt(6))^k)/(2*sqrt(6)) = {1, 2, 9, 28, 101}. 101 is odd prime, so a(3) = 101.

Mathematica

g[n_, k_] := ((1 + Sqrt[n])^k - (1 - Sqrt[n])^k)/(2Sqrt[n]);
Table[k = 3; While[! PrimeQ[Expand@g[2n, k]], k++]; Expand@g[2n, k], {n, 41}]

Prog

(PARI)
g(n, k) = ([0, 1; 2*n-1, 2]^k*[0; 1])[1, 1]
a(n) = for(k=3, oo, if(ispseudoprime(g(n, k)), return(g(n, k)))) \\ Jason Yuen, Apr 12 2025

Crossrefs

Cf. A000129, A002605, A015518, A063727, A002532, A083099, A015519, A003683, A002534, A083102, A015520, A091914, A079773, A161007, A099134.

Sequence in context: A329006 A176619 A230379 * A088270 A259496 A338525

Adjacent sequences: A292844 A292845 A292846 * A292848 A292849 A292850

Keyword

nonn

Author

XU Pingya, Sep 24 2017

Status

approved