A208292 -id:A208292 - cp's OEIS frontend

A208293 Numbers n such that (n^2+1)/26 is prime.

21, 31, 109, 125, 135, 151, 161, 229, 281, 291, 359, 369, 385, 525, 541, 551, 619, 629, 645, 671, 681, 749, 759, 801, 879, 941, 1009, 1019, 1035, 1149, 1165, 1175, 1399, 1425, 1435, 1529, 1539, 1555, 1565, 1581, 1669, 1685, 1695, 1799, 1851, 1919, 1945, 1971

Offset: 1

Wolfdieter Lang, Feb 27 2012

nonn

The corresponding primes (n^2+1)/26 are given in A208292(n).
a(n) is the smallest positive representative of the class of
  nontrivial solutions of the congruence x^2==1 (Modd A208292(n)), if n>=2. The trivial solution is the class with representative x=1, which also includes -1. For Modd n see a comment on A203571. For n=1: a(1) = 21 == 13 (Modd 17), and 13 is the smallest positive solution >1.
The unique class of nontrivial solutions of the congruence x^2==1 (Modd p), with p an odd prime, exists for any p of the form 4*k+1, given in A002144. Here a subset of these primes is covered, the ones for k=k(n)=(a(n)^2-25)/(4*26). These values are 4, 9, 114, 150, 175, 219, ...

			a(3)=109 because (109^2+1)/26 = 457 is prime.
  109 = sqrt(26*457-1) = sqrt(8*1485+1).

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A208292, A207337, A207339, A129307, A027862, A002731.

Select[Range[10000], PrimeQ[(#^2 + 1)/26] &] (* T. D. Noe, Feb 28 2012 *)

is(n)=isprime((n^2+1)/26) \\ Charles R Greathouse IV, May 22 2017

a(n) = sqrt(26*A208292(n)-1) = sqrt(8*A208294(n)+1), n>=1.

A208294 Triangular numbers T from A000217 such that (4*T+1)/13 is prime.

Original entry on oeis.org

55, 120, 1485, 1953, 2278, 2850, 3240, 6555, 9870, 10585, 16110, 17020, 18528, 34453, 36585, 37950, 47895, 49455, 52003, 56280, 57970, 70125, 72010, 80200, 96580, 110685, 127260, 129795, 133903, 165025, 169653, 172578, 244650

Offset: 1

Wolfdieter Lang, Feb 27 2012

nonn

The corresponding primes are gven in A208292, where equivalent formulations are found.

The indices of these triangular numbers are given by (A208293(n)-1)/2.

			a(2) = 120. m(2)= 31: 120 = T((31-1)/2) = T(15)=A000217(15). (4*120+1)/13 = 37 = A208292(2).

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A208292. A208293, A207337, A207339, A129307, A027862, A002731.

tri = # (# + 1)/2 & /@ Range@ 1000; Select[ tri, PrimeQ[(4 # + 1)/13] &] (* Robert G. Wilson v, Feb 28 2012 *)

a(n) = T(K(n)):= A000217(K(n)) with K(n)=(A208293(n)-1)/2.

cp's OEIS Frontend

A208293 Numbers n such that (n^2+1)/26 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula