A207337 -id:A207337 - cp's OEIS frontend

A207339 Triangular numbers T from A000217 such that (4*T+1)/5 is prime.

6, 21, 36, 66, 91, 136, 171, 351, 496, 561, 741, 946, 1176, 1326, 1596, 2016, 2346, 2701, 2926, 3321, 3486, 4851, 6216, 6441, 7626, 8646, 8911, 9591, 10011, 10296, 11026, 11476, 13041, 15051, 16471, 16836, 17391, 18336, 19701, 21736

Offset: 1

Wolfdieter Lang, Feb 27 2012

nonn

The corresponding primes are gven in A207337, where also equivalent formulations are found.

The indices of these triangular numbers are given by (A002733(n)-1)/2.

			a(3) = 36 = T((17-1)/2) = T(8)=A000217(8). (4*36+1)/5 = 29 = A207337(3).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A207337, A129307, A027862, A002731.

Select[Accumulate[Range[300]],PrimeQ[(4#+1)/5]&] (* Harvey P. Dale, Sep 18 2019 *)

a(n) = T(K(n)):= A000217(K(n)) with K(n)=(m(n)-1)/2, and m(n) given in A002733(n).

A002733 Numbers k such that (k^2 + 1)/10 is prime.

Original entry on oeis.org

7, 13, 17, 23, 27, 33, 37, 53, 63, 67, 77, 87, 97, 103, 113, 127, 137, 147, 153, 163, 167, 197, 223, 227, 247, 263, 267, 277, 283, 287, 297, 303, 323, 347, 363, 367, 373, 383, 397

Offset: 1

N. J. A. Sloane

nonn

Contribution from Wolfdieter Lang, Feb 27 2012: (Start)
The corresponding primes (n^2 + 1)/10 are given in A207337(n).
a(n) is the smallest positive representative of the class of nontrivial solutions of the congruence x^2 == 1 (Modd A207337(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) = 7 == 3 (Modd 5), and 3 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 - 9)/40. These k-values are [1, 4, 7, 13, 18, 27, 34, 70, 99, 112, ...].
(End)

L. Euler, De numeris primis valde magnis (E283), reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 3, p. 25.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
L. Euler, De numeris primis valde magnis (E283), The Euler Archive

Cf. A000196, A010051, A002522, A002731, A002732.

a002733 = a000196 . (subtract 1) . (* 10) . a207337
-- Reinhard Zumkeller, Apr 06 2012

a := [ ]: for n from 1 to 400 do if (n^2+1 mod 10) = 0 and isprime((n^2+1)/10) then a := [ op(a), n ]; fi; od;

Select[Range[573], PrimeQ[(#^2 + 1)/10] &] (* T. D. Noe, Feb 28 2012 *)

forstep(n=7,1e3,[6,4],if(isprime(n^2\10+1),print1(n", "))) \\ Charles R Greathouse IV, Mar 11 2012

a(n) = sqrt(10*A207337(n)-1) = sqrt(8*A207339(n)+1), n >= 1. - Wolfdieter Lang, Feb 27 2012

A208292 Primes of the form (n^2+1)/26.

Original entry on oeis.org

17, 37, 457, 601, 701, 877, 997, 2017, 3037, 3257, 4957, 5237, 5701, 10601, 11257, 11677, 14737, 15217, 16001, 17317, 17837, 21577, 22157, 24677, 29717, 34057, 39157, 39937, 41201, 50777, 52201, 53101, 75277, 78101, 79201, 89917, 91097, 93001, 94201, 96137

Offset: 1

Wolfdieter Lang, Feb 27 2012

nonn

Equivalently, primes of the form (K^2 + (K+1)^2)/13. The
connection to the primes of the form (m^2+1)/26 is given by m=2*K+1 (m is necessarily odd).
The corresponding m=m(n) values are given in A208293(n).
Equivalently, primes of the form (4*T(K)+1)/13, with the
  corresponding triangular numbers T(K):=A000217(K), for
  K=K(n)=(m(n)-1)/2, given in A208294(n).
For n>=2 the smallest positive representative of the class of
  nontrivial solutions of the congruence x^2==1 (Modd a(n)) is
  x=m(n). The trivial solution is the class with representative x=1, which also includes -1. For the prime
  a(1)=17 the nontrivial solution is 13 (see A002733(2)). Unique  nontrivial smallest positive representatives exist for the solutions  for any prime of the form 4*k+1, given in A002144. Here the subset with k=k(n)=(a(n)-1)/4 appears, namely 4,9,114,150,175,219,.... For Modd n see a comment on A203571.
These primes with corresponding m values are such that floor(m(n)^2/p(n)) = 5^2, n>=1.

			a(3)=457, m(3)=A208293(3)=109. T(K(3))=A000217((109-1)/2)=
  A000217(54)=A208294(3)=1485.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Cf. A207337, A207339 (case floor(m^2/p)=3^2); A129307, A027862, A002731 (case floor(m^2/p)=1^2).

Select[(Range[2000]^2 + 1)/26, PrimeQ] (* T. D. Noe, Feb 28 2012 *)

a(n) is the n-th member of the increasingly ordered list of primes of the form (m^2+1)/10, where m=m(n) is necessarily an odd integer, the positive one is A208293(n).

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

Original entry on oeis.org

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.

A154409 Primes of the form 10n^2+6n+1.

Original entry on oeis.org

17, 53, 109, 281, 397, 1061, 1277, 2341, 2657, 4973, 6917, 8009, 9181, 10433, 13177, 13913, 14669, 18749, 20521, 25301, 26317, 28409, 32833, 42641, 45293, 46649, 56701, 58217, 59753, 67733, 69389, 76213, 77969, 83357, 85193, 87049, 90821

Offset: 1

Vincenzo Librandi, Jan 09 2009

Subsequence of A207337. Primes of the form n^2+(3n+1)^2. - Bruno Berselli, Jul 17 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A017305, A154408.

[a: n in [0..200] | IsPrime(a) where a is 10*n^2+6*n+1]; // Vincenzo Librandi, Jul 17 2012

Select[Table[10n^2+6n+1,{n,0,700}],PrimeQ] (* Vincenzo Librandi, Jul 17 2012 *)

Misleading formula removed - R. J. Mathar, Oct 18 2010

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A207339 Triangular numbers T from A000217 such that (4*T+1)/5 is prime.

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A002733 Numbers k such that (k^2 + 1)/10 is prime.

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A208292 Primes of the form (n^2+1)/26.

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

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

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A154409 Primes of the form 10n^2+6n+1.

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