A027862 -id:A027862 - cp's OEIS frontend

A276261 Centered 21-gonal primes.

127, 211, 757, 2521, 2857, 6301, 8527, 16381, 19867, 23689, 24697, 27847, 32341, 37171, 38431, 42337, 66361, 68041, 82237, 89839, 97777, 103951, 114661, 140071, 152461, 162751, 170689, 192781, 204331, 216217, 231547, 240997, 284131, 308827, 353557, 357421, 385057, 389089

Offset: 1

Ilya Gutkovskiy, Aug 26 2016

nonn

Primes of the form (21*k^2 + 21*k + 2)/2.

Numbers k such that (21*k^2 + 21*k + 2)/2 is prime: 3, 4, 8, 15, 16, 24, 28, 39, 43, 47, 48, 51, 55, 059, 60, 63, 79, 80, 88, 92, 96, 99, ...

OEIS Wiki, Centered polygonal numbers
Eric Weisstein's World of Mathematics, Centered Polygonal Number
Index entries for sequences related to centered polygonal numbers

Cf. A000040, A069178.

Cf. similar sequences of the centered k-gonal primes: A125602 (k = 3), A027862 (k = 4), A145838 (k = 5), A002407 (k = 6), A144974 (k = 7), A090562 (k = 10), A262344 (k = 11), A262493 (k = 13), A264821 (k = 14), A264822 (k = 15), A264823 (k = 16), A264824 (k = 17), A264825 (k = 18), A264844 (k = 19), A264845 (k = 20), A201715 (k = 24).

Intersection[Table[(21 k^2 + 21 k + 2)/2, {k, 0, 1000}], Prime[Range[33000]]]

lista(nn) = for(n=1, nn, if(isprime(p=(21*n^2 + 21*n + 2)/2), print1(p, ", "))); \\ Altug Alkan, Aug 26 2016

A218207 Number of n-digit primes of the form (k-1)^2 + k^2.

Original entry on oeis.org

1, 3, 6, 16, 42, 107, 286, 764, 2124, 5917, 17250, 49818, 145429

Offset: 1

Martin Renner, Oct 23 2012

Cf. A027861, A027862, A218208, A218209, A218211.

n = 0; Table[cnt = 0; While[n++; p = 2*n^2 - 2*n + 1; p < 10^e, If[PrimeQ[p], cnt++]]; n--; cnt, {e, 10}] (* T. D. Noe, Oct 23 2012 *)

a(n) = A218208(n) - A218208(n-1)

A218208 Number of primes up to 10^n that are of the form (k-1)^2 + k^2.

Original entry on oeis.org

1, 4, 10, 26, 68, 175, 461, 1225, 3349, 9266, 26516, 76334, 221763

Offset: 1

Martin Renner, Oct 23 2012

Cf. A027861, A027862, A218207, A218210, A218212.

n = 0; cnt = 0; Table[While[n++; p = 2*n^2 - 2*n + 1; p < 10^e, If[PrimeQ[p], cnt++]]; n--; cnt, {e, 10}] (* T. D. Noe, Oct 23 2012 *)

a(n) = sum(A218207(k), k=1..n)

A376992 a(n) is the least n-digit prime of the form j^2 + (j+1)^2.

Original entry on oeis.org

5, 13, 113, 1013, 10513, 100801, 1006781, 10030721, 100040513, 1001057513, 10000515313, 100016728501, 1000078402181, 10000013617661, 100000472012281, 1000000064846161, 10000005481873013, 100000002459693601, 1000000116852093013, 10000000062611784481, 100000001234170737761

Offset: 1

Stefano Spezia, Oct 11 2024

Cf. A001844, A027861, A027862, A376907, A376993.

f:= proc(n) local j,x;
     for j from ceil((sqrt(2*10^(n-1)-1)-1)/2) do
       x:= j^2 + (j+1)^2;
       if isprime(x) then return x fi
     od
end proc:
map(f, [$1..40]); # Robert Israel, Oct 13 2024

a[n_]:=Module[{k=1}, While[!PrimeQ[m=2k^2+2k+1]||IntegerLength[m]

from math import isqrt
from itertools import count
from sympy import prime
def A376992(n):
    for k in count(isqrt(((a:=10**(n-1))<<1)-1>>2)):
        m = 2*k*(k+1)+1
        if m >= a and isprime(m):
            return m # Chai Wah Wu, Oct 13 2024

Conjecture: a(n+1)/a(n) ~ 10.

A076727 Primes of the form x^2 + (x+3)^2.

Original entry on oeis.org

17, 29, 89, 149, 269, 317, 617, 929, 1109, 1409, 2969, 3449, 3617, 4517, 5309, 6389, 7817, 8069, 8849, 12329, 14969, 17117, 17489, 21017, 23117, 23549, 27617, 30509, 32009, 33029, 34589, 35117, 41189, 42929, 43517, 47129, 48989, 52817, 60209

Offset: 1

Cino Hilliard, Oct 28 2002

Each prime of the form 4k+1 has a unique representation as x^2+y^2; these primes have y-x=3.

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

Cf. A002144, A027862.

[a: n in [0..200] | IsPrime(a) where a is 2*n^2+6*n+9]; // Vincenzo Librandi, Jul 15 2012

Select[Table[2*n^2+6*n+9,{n,0,300}],PrimeQ] (* Vincenzo Librandi, Jul 15 2012 *)

Edited by Don Reble, May 03 2006

A154428 Primes of the form 50n^2 + 10n + 1.

Original entry on oeis.org

61, 1301, 1861, 2521, 5101, 7321, 8581, 9941, 14621, 16381, 20201, 24421, 26681, 34061, 36721, 51521, 68821, 76441, 97241, 101701, 106261, 110921, 135721, 163021, 168781, 199081, 205441, 218461, 252761, 282001, 304981, 312841, 337021, 353641

Offset: 1

Vincenzo Librandi, Jan 09 2009

Subsequence of A027862 associated with the values of A027861 that are multiples of 5. [R. J. Mathar, Jan 12 2009]

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

Cf. A017281, A048161.

Filtered(List([1..100],n->50*n^2+10*n+1),IsPrime); # Muniru A Asiru, Apr 25 2019

[a: n in [0..100] | IsPrime(a) where a is 50*n^2 + 10*n + 1]; // Vincenzo Librandi, Jul 23 2012

select(isprime,[50*n^2+10*n+1$n=1..100])[]; # Muniru A Asiru, Apr 25 2019

Select[Table[50n^2+10n+1,{n,0,200}],PrimeQ] (* Vincenzo Librandi, Jul 23 2012 *)

for (n=0, 100, if (isprime (k=50*n^2+10*n+1), print1 (k, ", "))); \\ Vincenzo Librandi, Jul 23 2012

Replaced 13721 by 135721 - R. J. Mathar, Jan 12 2009

A166080 Nonprimes of the form (k^2+1)/2.

Original entry on oeis.org

1, 25, 85, 145, 221, 265, 365, 481, 545, 685, 841, 925, 1105, 1405, 1513, 1625, 1985, 2245, 2665, 2813, 2965, 3281, 3445, 3785, 3961, 4141, 4325, 4705, 4901, 5305, 5513, 5725, 5941, 6161, 6385, 6613, 6845, 7081, 7565, 7813, 8065, 8321, 8845, 9113, 9385

Offset: 1

Juri-Stepan Gerasimov, Oct 06 2009

Or, 1 together with composite numbers of the form i^2+(i+1)^2. See A012132. - N. J. A. Sloane, Feb 29 2020

A166080 U A027862 = A001844.

			a(1)=(1^2+1)/2=1; a(2)=(7^2+1)/2=25.

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

Cf. A001844, A027862, A012132, A141468.

Select[(Range[0,150]^2+1)/2,IntegerQ[#]&&!PrimeQ[#]&] (* Harvey P. Dale, Aug 09 2025 *)

a(n) = 2n^2 + O(n^2/log n). - Charles R Greathouse IV, Mar 21 2014

Replaced 6261 by 6161 - R. J. Mathar, Oct 07 2009

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

A276261 Centered 21-gonal primes.

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A218207 Number of n-digit primes of the form (k-1)^2 + k^2.

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A218208 Number of primes up to 10^n that are of the form (k-1)^2 + k^2.

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A376992 a(n) is the least n-digit prime of the form j^2 + (j+1)^2.

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A076727 Primes of the form x^2 + (x+3)^2.

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Magma

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A154428 Primes of the form 50n^2 + 10n + 1.

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A166080 Nonprimes of the form (k^2+1)/2.

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

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