A048161 -id:A048161 - cp's OEIS frontend

A005383 Primes p such that (p+1)/2 is prime.

3, 5, 13, 37, 61, 73, 157, 193, 277, 313, 397, 421, 457, 541, 613, 661, 673, 733, 757, 877, 997, 1093, 1153, 1201, 1213, 1237, 1321, 1381, 1453, 1621, 1657, 1753, 1873, 1933, 1993, 2017, 2137, 2341, 2473, 2557, 2593, 2797, 2857, 2917, 3061, 3217, 3253

Offset: 1

N. J. A. Sloane

Also, n such that sigma(n)/2 is prime. - Joseph L. Pe, Dec 10 2001; confirmed by Vladeta Jovovic, Dec 12 2002
Primes that are followed by twice a prime, i.e., are followed by a semiprime. (For primes followed by two semiprimes, see A036570.) - Zak Seidov, Aug 03 2013, Dec 31 2015
If A005382(n) is in A168421 then a(n) is a twin prime with a Ramanujan prime, A104272(k) = a(n) - 2. - John W. Nicholson, Jan 07 2016
Starting with 13 all terms are congruent to 1 mod 12. - Zak Seidov, Feb 16 2017
Numbers n such that both n and n+12 are terms are 61, 661, 1201, 4261, 5101, 6121, 6361 (all congruent to 1 mod 60). - Zak Seidov, Mar 16 2017
Primes p for which there exists a prime q < p such that 2q == 1 (mod p). Proof: q = (p + 1)/2. - David James Sycamore, Nov 10 2018
Prime numbers n such that phi(sigma(2n)) = phi(2n), excluding n=3 and n=5; as well as phi(sigma(3n)) = phi(3n), excluding n=3 only. - Richard R. Forberg, Dec 22 2020

			Both 3 and (3+1)/2 = 2 are primes, both 5 and (5+1)/2 = 3 are primes. - _Zak Seidov_, Nov 19 2012

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
R. P. Boas & N. J. A. Sloane, Correspondence, 1974
Benoit Cloitre, On the fractal behavior of primes, 2011.

Cf. A005382, A057326, A057327, A057328, A057329, A057330, A005603.
A subsequence of A000040 which has A036570 as subsequence.
Cf. A005385, A010051, A065091, A048161, A036570.

a005383 n = a005383_list !! (n-1)
a005383_list = [p | p <- a065091_list, a010051 ((p + 1) `div` 2) == 1]
-- Reinhard Zumkeller, Nov 06 2012

LIMIT = 8000 % Find all members of A005383 less than LIMIT A = primes(LIMIT); n = length(A); %n is number of primes less than LIMIT B = 2*A - 1; C = ones(n, 1)*A; %C is an n X n matrix, with C(i, j) = j-th prime D = B'*ones(1, n); %D is an n X n matrix, with D(i, j) = (i-th prime)*2 - 1 [i, j] = find(C == D); A(j)

[n: n in [1..3300] | IsPrime(n) and IsPrime((n+1) div 2) ]; // Vincenzo Librandi, Sep 25 2012

for n to 300 do
  X := ithprime(n);
Y := ithprime(n+1);
Z := 1/2 mod Y;
  if isprime(Z) then print(Y);
end if:
end do:
# David James Sycamore, Nov 11 2018

Select[Prime[Range[1000]], PrimeQ[(# + 1)/2] &] (* Zak Seidov, Nov 19 2012 *)

A005383_list(n) = select(m->isprime(m\2+1),primes(n)[2..n]) \\ Charles R Greathouse IV, Sep 25 2012

from sympy import isprime
[n for n in range(3, 5000) if isprime(n) and isprime((n + 1)//2)]
# Indranil Ghosh, Mar 17 2017

[n for n in prime_range(3, 1000) if is_prime((n + 1) // 2)]
# F. Chapoton, Dec 17 2019

a(n) = A129521(n)/A005382(n). - Reinhard Zumkeller, Apr 19 2007
A000035(a(n))*A010051(a(n))*A010051((a(n)+1)/2) = 1. - Reinhard Zumkeller, Nov 06 2012
a(n) = 2*A005382(n) - 1. - Zak Seidov, Nov 19 2012
a(n) = A005382(n) + phi(A005382(n)) = A005382(n) + A000010(A005382(n)). - Torlach Rush, Mar 10 2014

More terms from David Wasserman, Jan 18 2002

Name changed by Jianing Song, Nov 27 2021

A017281 a(n) = 10*n + 1.

Original entry on oeis.org

1, 11, 21, 31, 41, 51, 61, 71, 81, 91, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 201, 211, 221, 231, 241, 251, 261, 271, 281, 291, 301, 311, 321, 331, 341, 351, 361, 371, 381, 391, 401, 411, 421, 431, 441, 451, 461, 471, 481, 491, 501, 511, 521, 531

Offset: 0

N. J. A. Sloane

Equals [1, 2, 3, ...] convolved with [1, 9, 0, 0, 0, ...]. - Gary W. Adamson, May 30 2009
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=10, (i>1), A[i,i-1] = -1, and A[i,j]=0 otherwise. Then, for n>=2, a(n-1) = -coeff(charpoly(A,x),x^(n-1)). - Milan Janjic, Feb 21 2010
Positive integers with last decimal digit = 1. - Wesley Ivan Hurt, Jun 17 2015
Also the number of (not necessarily maximal) cliques in the 2n-crossed prism graph. - Eric W. Weisstein, Nov 29 2017
From Martin Renner, May 28 2024: (Start)
Also number of squares in a grid cross with equally long arms and a width of two points (cf. A017113), e.g. for n = 2 there are nine squares of size 1 unit of area, four of size 2, two of size 5, four of size 8 and two of size 13, thus a total of 21 squares.
    · ·           · ·           · ·           · ·           * ·
    · ·           · ·           * ·           * ·           · ·
* * · · · ·   · · * · · ·   · · · · * ·   · · · · · ·   · · · · · *
* * · · · ·   · * · * · ·   · * · · · ·   * · · · * ·   * · · · · ·
    · ·           * ·           · *           · ·           · ·
    · ·           · ·           · ·           * ·           · *
The possible areas of the squares are given by ceiling(k^2/2) for 1 <= k <= 2*n+1, cf. A000982. In general, there are 4*n + 1 squares with one unit area to be found in the cross, cf. A016813, for n > 0 always four squares of even area and two squares of odd area > 1. (End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Milan Janjic, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8.
Tanya Khovanova, Recursive Sequences
Eric Weisstein's World of Mathematics, Clique
Eric Weisstein's World of Mathematics, Crossed Prism Graph
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A093645 (column 1).
Subsequence of A034709, together with A017293, A017329, A139222, A139245, A139249, A139264, A139279 and A139280.
Cf. A048161, A154428.
Cf. A005408, A016813, A016921, A017533, A161700, A128470, A158057, A161705, A161709, A161714.
Cf. A030430 (primes).
Cf. A272914, first comment. [Bruno Berselli, May 26 2016]

List([0..60], n-> 10*n+1 ); # G. C. Greubel, Sep 18 2019

a017281 = (+ 1) . (* 10)
a017281_list = [1,11..]  -- Reinhard Zumkeller, Apr 16 2012

[10*n+1 : n in [0..60]]; // Zaki Khandaker, May 16 2015

A017281:=n->10*n + 1; seq(A017281(n), n=0..80); # Wesley Ivan Hurt, Jan 29 2014

f[n_] := FromDigits[IntegerDigits[n^2, n + 1]]; Array[f, 60] (* Robert G. Wilson v, Apr 14 2009 *)
Range[1, 1000, 10] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)
(* From Eric W. Weisstein, Nov 29 2017: (Start) *)
Table[10n+1, {n, 0, 60}]
10*Range[0, 60] + 1
LinearRecurrence[{2, -1}, {11, 21}, {0, 60}]
CoefficientList[Series[(1+9x)/(1-x)^2, {x, 0, 60}], x] (* End *)

Vec((1+9*x)/(1-x)^2 + O(x^80)) \\ Michel Marcus, Jun 17 2015

[10*n+1 for n in (0..60)] # G. C. Greubel, Sep 18 2019

G.f.: (1+9*x)/(1-x)^2.
a(n) = 20*n - a(n-1) - 8, with a(0)=1. - Vincenzo Librandi, Nov 20 2010
a(n) = 2*a(n-1) - a(n-2), for n > 2. - Wesley Ivan Hurt, Jun 17 2015
E.g.f.: (1 + 10*x)*exp(x). - G. C. Greubel, Sep 18 2019

A067756 Prime hypotenuses of Pythagorean triangles with a prime leg.

Original entry on oeis.org

5, 13, 61, 181, 421, 1741, 1861, 2521, 3121, 5101, 8581, 9661, 16381, 19801, 36721, 60901, 71821, 83641, 100801, 106261, 135721, 161881, 163021, 199081, 205441, 218461, 273061, 282001, 337021, 388081, 431521, 491041, 531481, 539761, 552301

Offset: 1

Henry Bottomley, Jan 31 2002

nonn

Apart from the first two terms, every term is congruent to 1 modulo 60 and is of the form 450k^2 +- 30k + 1 or 450k^2 +- 330k + 61 for some k.
Every term of the sequence after the second is a prime p congruent to 1 (mod 60), i.e., for n > 2, a(n) is a subsequence of A088955. The Pythagorean triple is {sqrt(2p-1), p-1, p}. - Lekraj Beedassy, Mar 12 2002
Primes p such that 2*p-1 is the square of a prime. - Robert Israel, Sep 16 2014
Primes p of the form ((q+1)/2)^2 + ((q-1)/2)^2, where q is a prime; then q belongs to A048161. - Thomas Ordowski, May 22 2015
The other (i.e., long) leg of the Pythagorean triangle is p-1. - Zak Seidov, Oct 30 2015

			For a(1)=5, the right triangle is 3, 4, 5 with 3 and 5 prime.
For a(10)=5101, the right triangle is 101, 5100, 5101 with 101 and 5101 prime.

Robert Israel, Table of n, a(n) for n = 1..10000 (first 184 terms from Andreas Boe)
H. Dubner and T. Forbes, Prime Pythagorean triangles, J. Integer Seqs., Vol. 4 (2001), #01.2.3.

Contains every value of A051859.

N:= 10^8: # to get all terms <= N
Primes:= select(isprime,[$3..floor(sqrt(2*N-1))]):
f:= proc(p) local q; q:= (p^2+1)/2; if isprime(q) then q else NULL fi end proc:
map(f, Primes); # Robert Israel, Sep 16 2014

f[n_]:=((p-1)/2)^2+((p+1)/2)^2; lst={};Do[p=Prime[n];If[PrimeQ[f[p]],AppendTo[lst,f[p]]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 27 2009 *)

forprime(p=3,10^3,if(isprime(q=(p^2+1)/2),print1(q,", "))) \\ Derek Orr, Apr 30 2015

a(n) = (A048161(n)^2 + 1)/2 = A067755(n) + 1.

A063440 Number of divisors of n-th triangular number.

Original entry on oeis.org

1, 2, 4, 4, 4, 4, 6, 9, 6, 4, 8, 8, 4, 8, 16, 8, 6, 6, 8, 16, 8, 4, 12, 18, 6, 8, 16, 8, 8, 8, 10, 20, 8, 8, 24, 12, 4, 8, 24, 12, 8, 8, 8, 24, 12, 4, 16, 24, 9, 12, 16, 8, 8, 16, 24, 24, 8, 4, 16, 16, 4, 12, 36, 24, 16, 8, 8, 16, 16, 8, 18, 18, 4, 12, 24, 16, 16, 8, 16, 40, 10, 4, 16

Offset: 1

Henry Bottomley, Jul 24 2001

a(n) = 4 iff either n is in A005383 or n/2 is in A005384.
a(n) is odd iff n is in A001108.
a(n) = 6 if either n = 18 or n = q^2 where q is in A048161 or n = 2 q^2 - 1 where q is in A106483. - Robert Israel, Oct 26 2015
From Bernard Schott, Aug 29 2020: (Start)
a(n-1) is the number of solutions in positive integers (x, y, z) to the simultaneous equations (x + y - z = n, x^2 + y^2 - z^2 = n) for n > 1. See the British Mathematical Olympiad link. In this case, one always has z > x and z > y.
For n = 12 as in the Olympiad problem, the a(11) = 8 solutions are (13,78,79), (14,45,47), (15,34,37), (18,23,29), (23,18,29), (34,15,37), (45,14,47), (78,13,79). (End)

			a(6) = 4 since 1+2+3+4+5+6 = 21 has four divisors {1,3,7,21}.

Steve Dinh, The Hard Mathematical Olympiad Problems And Their Solutions, AuthorHouse, 2011, Problem 2 of the British Mathematical Olympiad 2007, page 28.

Ray Chandler, Table of n, a(n) for n = 1..10000
British Mathematical Olympiad 2007/2008, Round 1, Problem 2.
Index to sequences related to Olympiads.

Cf. A000005, A000217.

Cf. A001108, A005383, A005384, A048161, A060778, A081978 (greedy inverse), A106483, A101755 (indices of records), A101756 (records).

seq(numtheory:-tau(n*(n+1)/2), n=1..100); # Robert Israel, Oct 26 2015

DivisorSigma[0,#]&/@Accumulate[Range[90]] (* Harvey P. Dale, Apr 15 2019 *)

for (n=1, 10000, write("b063440.txt", n, " ", numdiv(n*(n + 1)/2)) ) \\ Harry J. Smith, Aug 21 2009

a(n)=factorback(apply(numdiv,if(n%2,[n,(n+1)/2],[n/2,n+1]))) \\ Charles R Greathouse IV, Dec 27 2014

vector(100, n, numdiv(n*(n+1)/2)) \\ Altug Alkan, Oct 26 2015

a(n) = A000005(A000217(n)).
From Robert Israel, Oct 26 2015: (Start)
a(2k) = A000005(k)*A000005(2k+1).
a(2k+1) = A000005(2k+1)*A000005(k+1).
gcd(a(2k), a(2k+1)) = A000005(2k+1) * A060778(k). (End)

A109953 Primes p such that p^2+2 is a semiprime.

Original entry on oeis.org

2, 7, 11, 17, 29, 37, 43, 53, 73, 79, 83, 97, 137, 191, 233, 251, 263, 269, 271, 277, 281, 359, 379, 389, 433, 461, 479, 521, 541, 577, 601, 631, 647, 677, 691, 719, 739, 827, 829, 863, 881, 929, 947, 983, 997, 1033, 1063, 1087, 1109, 1187, 1223

Offset: 1

Zak Seidov, Jul 06 2005

nonn

Cf. A048161 Primes p such that p^2+1 is a semiprime.

Primes p such that (p^2+2)/3 is prime. For all primes q>3, we have q=6k+-1 for some k, which makes it easy to show that 3 divides q^2+2. Hence if q^2+2 is a semiprime then (q^2+2)/3 must be prime. - T. D. Noe, May 05 2006

			a(2) = 7 is o.k. because 7^2+2=51=3*17 (semiprime), and 17 = A289135(2).

Harvey P. Dale, Table of n, a(n) for n = 1..2500

Cf. A048161, A289135.

Cf. A118915 (primes p such that (p^2+5)/6 is prime).

A109953=Select[Prime[Range[200]], Plus@@Last/@FactorInteger[ #^2+2]==2&]
Select[Prime[Range[200]],PrimeOmega[#^2+2]==2&] (* Harvey P. Dale, Nov 19 2011 *)

a(n) = sqrt(3*A289135(n) - 2). See the T. D. Noe comment above. - Wolfdieter Lang, Jul 19 2017

A057538 Birthday set of order 5: numbers congruent to +-1 modulo 2, 3, 4 and 5.

Original entry on oeis.org

1, 11, 19, 29, 31, 41, 49, 59, 61, 71, 79, 89, 91, 101, 109, 119, 121, 131, 139, 149, 151, 161, 169, 179, 181, 191, 199, 209, 211, 221, 229, 239, 241, 251, 259, 269, 271, 281, 289, 299, 301, 311, 319, 329, 331, 341, 349, 359, 361, 371, 379, 389, 391, 401, 409

Offset: 1

Andrew R. Feist (andrewf(AT)math.duke.edu), Sep 06 2000

Also numbers congruent to +-1 or +-11 modulo 30 and numbers k where (k^2 - 1)/120 is an integer; all but the first two prime legs of Pythagorean triangles which also have prime hypotenuses appear within in this sequence (A048161). - Henry Bottomley, Jan 31 2002
Numbers k such that k^2 == 1 (mod 30). - Gary Detlefs, Apr 16 2012
Subsequence of primes gives A045468. - Ray Chandler, Jul 29 2019

			229 is congruent to 1 (mod 2), 1 (mod 3), 1 (mod 4) and -1 (mod 5).
x+ 11*x^2 + 19*x^3 + 29*x^4 + 31*x^5 + 41*x^6 + 49*x^7 + 59*x^8 + 61*x^9 + ...

Michael De Vlieger, Table of n, a(n) for n = 1..10000
A. Feist, On the Density of Birthday Sets, The Pentagon, 60 (No. 1, Fall 2000), 31-35.
A. Feist, Maple source for birthday sets. [Broken link]
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A007310, A045468, A057539, A057540, A057541, A093722.

for n from 1 to 409 do if (n^2 mod 30 =1) then print(n) fi od; # Gary Detlefs, Apr 17 2012

a057538[n_] := Block[{f},
  f[x_] :=
   If[Mod[x, #] == 1 || Mod[x, #] == # - 1, True, False] & /@
    Range[2, 5];
Select[Range[n], DeleteDuplicates[f[#]] == {True} &]]; a057538[409] (* Michael De Vlieger, Dec 26 2014 *)

{a(n+1) = (n\4*3 + n%4)*10 + (-1)^(n\2)} /* Michael Somos, Oct 17 2006 */

A093722(n) = (a(n)^2 - 1)/120.
G.f.: x * (1 + 10*x + 8*x^2 + 10*x^3 + x^4) / ((1 - x) * (1 - x^4)). a(-1 - n) = -a(n). - Michael Somos, Jan 21 2012
4*a(n) = 30*(n+1) - 45 + 5*(-1)^n + 6*(-1)^floor((n+1)/2). - R. J. Mathar, Jul 30 2019

Corrected by Henry Bottomley, Jan 31 2002

Offset corrected to 1 by Ray Chandler, Jul 29 2019

A340480 Primes p such that (p^8 + 1)/2 is prime.

Original entry on oeis.org

13, 43, 47, 53, 239, 373, 409, 433, 491, 557, 577, 859, 1021, 1103, 1307, 1531, 1699, 1753, 1777, 1871, 2053, 2083, 2297, 2467, 2503, 2593, 2797, 2957, 3251, 3307, 3323, 3511, 3613, 4099, 4523, 4637, 4951, 4999, 5591, 5657, 5693, 5801, 5827, 5849, 6043, 6163

Offset: 1

Jon E. Schoenfield, Jan 31 2021

nonn

Expressions of the form m^j + 1 can be factored (e.g., m^3 + 1 = (m + 1)*(m^2 - m + 1)) for any positive integer j except when j is a power of 2, so (p^j + 1)/2 for prime p cannot be prime unless j is a power of 2. A005383, A048161, A176116, and this sequence list primes of the form (p^j + 1)/2 for j=2^0=1, j=2^1=2, j=2^2=4, and j=2^3=8, respectively.

(p^8 + 1)/2 is divisible by 17 when m mod 34 is 3, 5, 7, 11, 23, 27, 29, or 31.

			(3^8 + 1)/2 = 3281 = 17*193, so 3 is not a term.
(13^8 + 1)/2 = 407865361 is prime, so 13 is a term.
(17^8 + 1)/2 = 3487878721 = 18913 * 184417, so 17 is not a term.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Primes p such that (p^(2^k) + 1)/2 is prime: A005383 (k=0), A048161 (k=1), A176116 (k=2), (this sequence) (k=3).

Prime[Position[Table[(Prime[p]^8 + 1)/2, {p, 1, 803}], Integer?PrimeQ]] // Flatten (* _Robert P. P. McKone, Jan 31 2021 *)

isok(p) = isprime(p) && (p>2) && isprime((p^8 + 1)/2); \\ Michel Marcus, Feb 01 2021

A067755 Even legs of Pythagorean triangles whose other leg and hypotenuse are both prime.

Original entry on oeis.org

4, 12, 60, 180, 420, 1740, 1860, 2520, 3120, 5100, 8580, 9660, 16380, 19800, 36720, 60900, 71820, 83640, 100800, 106260, 135720, 161880, 163020, 199080, 205440, 218460, 273060, 282000, 337020, 388080, 431520, 491040, 531480, 539760, 552300

Offset: 1

Henry Bottomley, Jan 31 2002

nonn

Apart from the first two terms, every term is divisible by 60 and is of the form 450*k^2 +/- 30*k or 450*k^2 +/- 330*k + 60 for some k.

In such a triangle, this even leg is always the longer leg, and the hypotenuse = a(n) + 1. The Pythagorean triples are (A048161(n), a(n), A067756(n)), so, for a(2) = 12, the corresponding Pythagorean triple is (5, 12, 13). - Bernard Schott, Apr 12 2023

			4 is a term: in the right triangle (3, 4, 5), 3 and 5 are prime.
5100 is a term: in the right triangle (101, 5100, 5101), 101 and 5101 are prime.

Ray Chandler, Table of n, a(n) for n = 1..10000
H. Dubner and T. Forbes, Prime Pythagorean triangles, J. Integer Seqs., Vol. 4 (2001), #01.2.3.

Cf. A048161, A067756. Contains every value of A051858.

lst={}; Do[q=(Prime[n]^2+1)/2; If[PrimeQ[q], AppendTo[lst, (Prime[n]^2-1)/2]], {n, 200}]; lst (* Frank M Jackson, Nov 02 2013 *)

a(n) = (A048161(n)^2 - 1)/2 = A067756(n) - 1.

A068501 Values m such that the consecutive pair parameters(m,m+1) generate Pythagorean triples whose odd terms are both prime.

Original entry on oeis.org

1, 2, 5, 9, 14, 29, 30, 35, 39, 50, 65, 69, 90, 99, 135, 174, 189, 204, 224, 230, 260, 284, 285, 315, 320, 330, 369, 375, 410, 440, 464, 495, 515, 519, 525, 534, 545, 564, 575, 585, 590, 680, 719, 729, 744, 749, 765, 854, 870, 905, 915, 950, 974, 1080, 1119

Offset: 1

Lekraj Beedassy, Mar 25 2002

nonn

Setting u=m; v=m+1, triples (a,b,c) with a=u+v, b=2*u*v, c = u^2+v^2 = (a^2+1)/2 correspond to (A048161, A067755, A067756), a and c being both prime.

Zak Seidov, Table of n, a(n) for n = 1..10000
Robert Simms, Deriving Pythagorean Triples (web archive)

Cf. A051892.

lst={};Do[If[PrimeQ[(n+1)^2-n^2]&&PrimeQ[(n+1)^2+n^2],AppendTo[lst,n]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 01 2010 *)
Reap[Do[a=Prime[k];If[PrimeQ[(a^2+1)/2],Sow[(a-1)/2]],{k,2,10^5}]][[2,1]](* Zak Seidov, Apr 16 2011 *)

More terms from Larry Reeves (larryr(AT)acm.org), Jun 19 2002

A048270 Sequence of 2 Pythagorean triangles, each with a leg and hypotenuse prime. The leg of the second triangle is the hypotenuse of the first.

Original entry on oeis.org

3, 11, 19, 59, 271, 349, 521, 929, 1031, 1051, 1171, 2381, 2671, 2711, 2719, 3001, 3499, 3691, 4349, 4691, 4801, 4999, 5591, 5669, 6101, 6359, 6361, 7159, 7211, 7489, 8231, 8431, 8761, 9241, 10099, 10139, 11719, 11821, 12239, 12281, 12781

Offset: 1

Harvey Dubner (harvey(AT)dubner.com)

nonn

It is conjectured that there are infinitely many such pairs of triangles.

Subsequence of A048161. - Lekraj Beedassy, Sep 16 2005

			p(1)=3 because 3 is prime, 5 = (3*3 + 1)/2 and 13 = (5*5 + 1)/2, 5, 13 both prime.

Ray Chandler, Table of n, a(n) for n = 1..10000
H. Dubner and T. Forbes, Prime Pythagorean triangles, J. Integer Seqs., Vol. 4 (2001), #01.2.3.

Cf. A048161, A048295, A308635, A308636. Primes in A116945.

For each p(n), there is a q=(p*p+1)/2 and r=(q*q+1)/2 such that p, q, r are all prime.

More terms from Ray Chandler, Jun 12 2019

cp's OEIS Frontend

A005383 Primes p such that (p+1)/2 is prime.

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A017281 a(n) = 10*n + 1.

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A067756 Prime hypotenuses of Pythagorean triangles with a prime leg.

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A063440 Number of divisors of n-th triangular number.

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PARI

PARI

PARI

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A109953 Primes p such that p^2+2 is a semiprime.

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A057538 Birthday set of order 5: numbers congruent to +-1 modulo 2, 3, 4 and 5.

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