A002731 -id:A002731 - cp's OEIS frontend

A091277 Numbers k such that the k-th prime is of the form m^2 + (m+1)^2.

3, 6, 13, 18, 30, 42, 65, 82, 112, 135, 170, 197, 212, 271, 284, 319, 353, 369, 445, 505, 612, 682, 933, 1069, 1193, 1226, 1286, 1510, 1609, 1676, 1711, 1789, 1900, 2161, 2241, 2285, 2363, 2450, 2667, 2712, 2924, 3247, 3644, 3894, 4000, 4100, 4367, 4647, 4922

Offset: 1

Ray Chandler, Jan 03 2004

Also, the k-th prime is of the form (m^2+1)/2.

Jinyuan Wang, Table of n, a(n) for n = 1..1000 (terms 1..500 from Harvey P. Dale)

A027862 indexed by A000040.

Cf. A002731, A027861, A027862.

Select[Range[5000],OddQ[Sqrt[2Prime[#]-1]]&] (* Harvey P. Dale, Jun 25 2018 *)

a(n)=k such that A000040(k)=A027862(n).

Offset changed to 1 by Jinyuan Wang, Aug 02 2021

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

A183064 Numbers k such that k^2+1 = 2*p^2, p prime.

Original entry on oeis.org

7, 41, 8119, 47321, 63018038201, 2470433131948081, 96845919575610633161, 19175002942688032928599, 5834531641231893991002972081099601, 6733044458057842709277507685523012161, 228725309250740208744750893347264645481

Offset: 1

Michel Lagneau, Feb 01 2011

nonn

Subset of A002315 (Numbers k such that k^2 + 1 = 2*q^2).

			a(2) = 41 because 41^2+1 = 2*29^2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..22

Cf. A002315, A005574, A002731, A174492.

with(numtheory):for n from 1 to 1000000 do : p:=ithprime(n):x:=2*p^2: y:=sqrt(x-1):if
  y=floor(y) then print(y):else fi:od:

list(lim)=my(v=List(),w=3+quadgen(32),k,n); while((k=imag((1+w)*w^n++))<=lim, if(ispseudoprime(sqrtint((k^2+1)/2)), listput(v,k))); Vec(v) \\ Charles R Greathouse IV, Sep 14 2015

More terms from Charles R Greathouse IV, Feb 01 2011

A181413 a(n) is the smallest number such that a(n)^2 + 1 is divisible by A002144(1)* A002144(2)... A002144(n).

Original entry on oeis.org

2, 8, 47, 2163, 18543, 241727, 3101272, 842894268, 8245041748, 521781374353, 101476250977928, 671795954794788, 32126984574675193, 425090834074746637, 309609468228403885693, 25836182225971546313682, 38544366727563360743217, 217758730168965028986551783, 25789605237863389220212237968, 309600287787935978580674202007

Offset: 1

Michel Lagneau, Jan 28 2011

nonn

			a(1) = 2 because A002144(1) | 2^2+1 = 5 ;
a(2)=8 because A002144(1) * A002144(2) | 8^2+1 = 5*13 ;
a(6) = 241727 because A002144(1) * A002144(2)*...* A002144(6) | 241727^2+1
  = 2 * 5 * 13 * 17 * 29 * 37 * 41 * 601.

Cf. A002144 (Pythagorean primes: primes of form 4n+1) A002731.

with(numtheory):nn:=1000:T:=array(1..1000):k:=1:for x from 1 to nn do: p:=4*x+1:if
  type(p, prime)=true then T[k]:=p:k:=k+1:else fi:od:pr:=1:for n from 1 to k do:
  pp:=pr*T[n] :ind:=0:for q from 1 to pp while (ind=0) do: z:=q^2+1:if irem(z,pp)=0
  and ind = 0 then ind: = 1:pr:=pp:print( q):else fi:od:od:
# Alternative
PP:= select(isprime, [seq(i,i=5..200,4)]):
f:= n -> min(map(t -> rhs(op(t)),[msolve(x^2+1, convert(PP[1..n],`*`))])):
map(f, [$1..20]); # Robert Israel, Feb 01 2019

More terms from Robert Israel, Feb 01 2019

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.

A284036 Positive integers n such that (n^2 - 3)/2 and (n^2 + 1)/2 are twin primes.

Original entry on oeis.org

3, 5, 11, 19, 25, 29, 65, 79, 101, 205, 209, 221, 245, 275, 289, 299, 349, 371, 409, 415, 449, 521, 535, 569, 571, 575, 595, 649, 661, 695, 739, 781, 791, 935, 949, 991, 1081, 1091, 1099, 1129, 1181, 1225, 1241, 1285, 1345, 1349, 1459, 1489, 1531, 1541, 1615

Offset: 1

Giuseppe Coppoletta, Mar 27 2017

nonn

All terms are obviously odd.

			25 is a term because (25^2 - 3)/2 = 311 and (25^2 + 1)/2 = 313 are twin primes.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002731, A109358, A048161, A110589, A284034.

filter:= n -> isprime((n^2-3)/2) and isprime((n^2+1)/2):
select(filter, [seq(i,i=1..2000,2)]); # Robert Israel, Apr 24 2017

Select[Range[1, 1285, 2], Times @@ Boole@ Map[PrimeQ, (#^2 + {-3, 1})/2] == 1 &] (* Michael De Vlieger, Mar 28 2017 *)

isok(n) = isprime((n^2 - 3)/2) && isprime((n^2 + 1)/2); \\ Michel Marcus, Apr 04 2017

from sympy import isprime
print([n for n in range(3, 1700, 2) if isprime((n**2 - 3)//2) and isprime((n**2 + 1)//2)]) # Indranil Ghosh, Apr 04 2017

[n for n in range(3,1700,2) if is_prime((n^2 - 3)//2) and is_prime((n^2 + 1)//2)]

A181619 Numbers k such that k^2+1 = 2p,(k+1)^2+1 = 5q, (k+2)^2+1 = 10r where p, q, and r are primes.

Original entry on oeis.org

11, 51, 61, 101, 221, 261, 571, 2271, 2821, 2871, 5071, 5651, 5761, 6561, 6951, 9751, 10461, 10851, 11621, 11711, 14961, 15911, 16551, 17171, 17601, 18511, 19071, 19551, 23151, 25261, 27351, 27751

Offset: 1

Michel Lagneau, Jan 31 2011

nonn

a(n) == 1 (mod 10).

			a(2) = 51 because 51^2+1 = 2*1301, 52^2+1 = 5*541, 53^2+1 = 10*281.

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

Cf. A002144, A005574, A002731.

with(numtheory):for n from 1 to 30000 do : x:=n^2+1:y:=(n+1)^2+1:z:=(n+2)^2+1:x1:=factorset(x):y1:=factorset(y):z1:=factorset(z):n1:=bigomega(x):n2:=bigomega(y):n3:=bigomega(z):if
  x1[1]=2 and n1=2 and y1[1]=5 and n2 = 2 and z1[1]=2 and z1[2]=5 and n3=3 then
  printf(`%d, `, n):else fi:od:

ksQ[k_]:=And@@PrimeQ[{(k^2+1)/2,((k+1)^2+1)/5,((k+2)^2+1)/10}]; Select[ Range[30000],ksQ] (* Harvey P. Dale, Sep 01 2013 *)

forstep(k=1,1e5,10,if(isprime(k^2\2+1)&isprime((k+1)^2\5+1)&isprime((k+2)^2\10+1),print1(k", ")))

A216091 Numbers n such that k == k^(q-1) mod q for k = 1, 2, ..., q-1, where q = n^2+1.

Original entry on oeis.org

1, 3, 5, 9, 11, 15, 19, 25, 29, 35, 39, 45, 47, 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

Offset: 1

Michel Lagneau, Sep 01 2012

nonn

It is interesting to note that this sequence is identical to A002731 except for the numbers 1 and 47. For instance, a(13) = 47 but (47^2+1)/2 = 1105 is not prime, but 47^2+1 = 2210 => k^2209 == {1, 2, 3, ..., 2208, 2209} mod 2210 for k = {1, 2, ..., 2210}.

Conclusion: the two numbers of this sequence 1, 47 are not in A002731. Are there other numbers?

			3 is in the sequence because, for q = 3^2 + 1 = 10 we obtain the congruences:
1^9 = 1 == 1 mod 10;
2^9 = 512 == 2 mod 10;
3^9 = 19683 == 3 mod 10;
4^9 = 262144 == 4 mod 10;
5^9 = 1953125 == 5 mod 10;
6^9 = 10077696 == 6 mod 10,
7^9 = 40353607 == 7 mod 10;
8^9 = 134217728 == 8 mod 10;
9^9 = 387420489 == 9 mod 10.

Cf. A002731.

with(numtheory):for n from 1  by 2 to 500 do:q:=n^2+1:if type(x,prime)=false then j:=0:for i from 1 to q do: if irem(i^(q-1),q)=i then j:=j+1:else fi:od:if j=q-1 then printf(`%d, `, n):else fi:fi:od:

f[n_] := Module[{q = n^2 + 1}, And @@ Table[PowerMod[k, q - 1, q] == k, {k, q - 1}]]; Select[Range[345], f] (* T. D. Noe, Sep 03 2012 *)

cp's OEIS Frontend

A091277 Numbers k such that the k-th prime is of the form m^2 + (m+1)^2.

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

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A183064 Numbers k such that k^2+1 = 2*p^2, p prime.

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A181413 a(n) is the smallest number such that a(n)^2 + 1 is divisible by A002144(1)* A002144(2)*...* A002144(n).

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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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A181413 a(n) is the smallest number such that a(n)^2 + 1 is divisible by A002144(1)* A002144(2)... A002144(n).