A129307 -id:A129307 - cp's OEIS frontend

A176796 Numbers k such that A129307(k) + A129307(k+1) is a square.

1, 3, 12, 14, 17, 25, 30, 35, 39, 69, 71, 74, 80, 83, 88, 102, 107, 122, 126, 129, 134, 151, 170, 172, 176, 184, 187, 202, 220, 239, 244, 249, 258, 261, 263, 272, 280, 283, 289, 298, 308, 321, 363, 371, 377, 386, 390, 403, 421, 432, 438, 447, 451, 453, 477, 480

Offset: 1

Giovanni Teofilatto, Apr 26 2010

nonn

Squares are in A075577.

Cf. A000217, A027861, A075577, A129307.

A027861 := proc(n) option remember; local a; if n= 1 then 1; else for a from procname(n-1)+1 do if isprime(a^2+(a+1)^2) then return a; end if; end do: end if; end proc:
A129307 := proc(n) A000217(A027861(n)) ; end proc:
A176796 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do if issqr(A129307(a)+A129307(a+1)) then return a; end if; end do: end if; end proc:
seq(A176796(n),n=1..80) ; # R. J. Mathar, Jun 28 2010

Extended beyond a(5) by R. J. Mathar, Jun 28 2010

A002731 Numbers k such that (k^2 + 1)/2 is prime.

Original entry on oeis.org

3, 5, 9, 11, 15, 19, 25, 29, 35, 39, 45, 49, 51, 59, 61, 65, 69, 71, 79, 85, 95, 101, 121, 131, 139, 141, 145, 159, 165, 169, 171, 175, 181, 195, 199, 201, 205, 209, 219, 221, 231, 245, 261, 271, 275, 279, 289, 299, 309, 315, 321, 325, 329, 335, 345, 349, 371, 375, 379, 391, 399, 405

Offset: 1

N. J. A. Sloane

From Wolfdieter Lang, Feb 24 2012: (Start)
a(n) = sqrt(8*A129307(n)+1) = sqrt(2*A027862(n)-1), n >= 1.
a(n) is the nontrivial solution of the congruence a(n)^2 == 1 (Modd A027862(n)). The trivial one is +1. For Modd n see a comment on A203571. E.g., a(3)^2 = 81 == 1 (Modd 41), see a comment on A027862.
(End)

L. Euler, De numeris primis valde magnis (E283), reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 3, p. 24.
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).

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
L. Euler, De numeris primis valde magnis (E283), The Euler Archive.

Cf. A027861. A027862 gives primes, A091277 gives prime indices.

Cf. A000982, A010051, A116945, A129307, A188546, A188547, A187431.

a002731 n = a002731_list !! (n-1)
a002731_list = filter ((== 1) . a010051 . a000982) [1, 3 ..]
-- Reinhard Zumkeller, Jul 13 2014

[n: n in [3..410] | IsPrime((n^2+1) div 2) ]; // Vincenzo Librandi, Sep 25 2012

Select[Range[400], PrimeQ[(#^2 + 1)/2] &] (* Alonso del Arte, Feb 24 2012 *)

forstep(n=1,10^3,2, if(isprime((n^2+1)/2),print1(n,", ")));
/* Joerg Arndt, Sep 02 2012 */

a(n) = 2*A027861(n) + 1.

A207337 Primes of the form (m^2+1)/10.

Original entry on oeis.org

5, 17, 29, 53, 73, 109, 137, 281, 397, 449, 593, 757, 941, 1061, 1277, 1613, 1877, 2161, 2341, 2657, 2789, 3881, 4973, 5153, 6101, 6917, 7129, 7673, 8009, 8237, 8821, 9181, 10433, 12041, 13177, 13469, 13913, 14669, 15761, 17389, 18233, 18749

Offset: 1

Wolfdieter Lang, Feb 27 2012

nonn

Equivalently, primes of the form (K^2 + (K+1)^2)/5. The connection to the primes of the form (m^2+1)/10 is given by m=2*K+1 (m is necessarily odd). The corresponsding m=m(n) values are given in A002733(n).
Equivalently, primes of the form (4*T(K)+1)/5, with the corresponding triangular numbers T(K):=A000217(K), for K(n)=(m(n)-1)/2, given in A207339(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)=5 the smallest positive nontrivial solution is 3 (see A027862(1) with  A002731(1)). Such a nontrivial smallest positive representative exists for each unique class of solutions of this congruence Modd p for any prime p of the form 4*k+1, given in A002144. Here the subset with k=k(n)=(a(n)-1)/4 appears, namely 1, 4, 7, 13, 18, 27, 34, 70,... For Modd n see a comment on A203571.

			a(3)=29, m(3)=A002733(3)=17. T(K(3))=A000217((17-1)/2)= A000217(8)=A207339(3)=36. (8^2+9^2)/5 = 29 = (4*36+1)/5.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A207339, A129307, A027862, A002731.

Cf. A010051, A002522.

a207337 n = a207337_list !! (n-1)
a207337_list = f a002522_list where
   f (x:xs) | m == 0 && a010051 y == 1 = y : f xs
            | otherwise                = f xs
            where (y,m) = divMod x 10
-- Reinhard Zumkeller, Apr 06 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, namely A002733(n).

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

Original entry on oeis.org

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).

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.

cp's OEIS Frontend

A176796 Numbers k such that A129307(k) + A129307(k+1) is a square.

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A002731 Numbers k such that (k^2 + 1)/2 is prime.

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A207337 Primes of the form (m^2+1)/10.

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