A056561 -id:A056561 - cp's OEIS frontend

A215697 Primes p = 2x + 1 such that x^2 + x + 41 is prime.

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 97, 101, 103, 107, 109, 127, 137, 139, 149, 151, 157, 167, 173, 181, 191, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 283, 293, 307, 313, 317, 331

Offset: 1

Pierre CAMI, Aug 21 2012

nonn

			17 is in the sequence because, given x = (17 - 1)/2 = 8, we have 8^2 + 8 + 41 = 113 which is prime.

Pierre CAMI, Table of n, a(n) for n = 1..10000

Cf. A056561, A005846.

Select[Prime[Range[100]], PrimeQ[((# - 1)/2)^2 + ((# - 1)/2) + 41] &] (* Alonso del Arte, Aug 21 2012 *)

for(x=1, 1e3, if(ispseudoprime(2*x+1), if(ispseudoprime(x^2+x+41), print1(2*x+1, ", ")))) \\ Felix Fröhlich, Aug 16 2014

A319906 Number of prime numbers of the form k^2 + k + 41 below 10^n.

Original entry on oeis.org

0, 8, 31, 86, 221, 581, 1503, 4149, 11355, 31985, 90940, 261081, 756081, 2208197, 6483148, 19132652, 56714624, 168806741, 504209234

Offset: 1

Amiram Eldar, Oct 01 2018

			The first 8 values of k^2 + k + 41 for k = 0 to 7 are above 10 and below 100: 41, 43, 47, 53, 61, 71, 83, 97, thus a(1) = 0 and a(2) = 8.

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.

Justin DeBenedetto and Jeremy Rouse, A 60,000 digit prime number of the form x^2 + x + 41, arXiv preprint arXiv:1207.7291 [math.NT] (2012).
Henri Cohen, High-precision computation of Hardy-Littlewood constants, (1998).
Henri Cohen, High-precision calculation of Hardy-Littlewood constants. [local copy with permission]
Gilbert W. Fung and Hugh C. Williams, Quadratic polynomials which have a high density of prime values, Mathematics of Computation, Vol. 55, No. 191 (1990), pp. 345-353.
G. H. Hardy and J. E. Littlewood, Some problems in "Partitio numerorum", III: On the expression of a number as a sum of primes, Acta Mathematica, Vol. 44 (1923), pp. 1-70.
R. A. Mollin, Prime-Producing Quadratics, The American Mathematical Monthly, Vol. 104, No. 6 (1997), pp. 529-544.
Daniel Shanks, Calculation and applications of Epstein zeta functions, Mathematics of Computation, Vol. 29, No. 129 (1975), pp. 271—287.
M. L. Stein, S. M. Ulam and M. B. Wells, A Visual Display of Some Properties of the Distribution of Primes, The American Mathematical Monthly, Vol. 71, No. 5 (1964), pp. 516-520.
Wikipedia, Ulam spiral.

Cf. A002837, A005846, A056561, A202018, A221712, A331876.

f[n_] := n^2 + n + 41; c = 0; k = 0; a={}; Do[f1 = f[k]; While[f1 < 10^n, If[PrimeQ[f1], c++]; k++; f1 = f[k]];  AppendTo[a, c], {n, 1, 10}]; a

According to Hardy and Littlewood's Conjecture F: a(n) ~ 2 * C * 10^(n/2)/(n*log(10)), where C = 3.319773... (Hardy-Littlewood constant for x^2+x+41, A221712).

A139220 Numbers k such that 41+(k+k^2)/2 = 41+A000217(k) is prime.

Original entry on oeis.org

0, 3, 11, 20, 23, 27, 32, 39, 44, 48, 51, 56, 59, 60, 83, 104, 108, 111, 116, 128, 132, 135, 143, 171, 188, 203, 207, 212, 227, 240, 251, 263, 275, 296, 300, 312, 315, 324, 356, 359, 363, 380, 384, 392, 399, 408, 443, 447, 476, 479, 483, 504, 507, 515, 527, 528

Offset: 1

Zak Seidov, Apr 11 2008

nonn

Corresponding values of primes are in A139219.

Numbers k such that both 41+(k+k^2)/2 and 41+(k+k^2) are primes, are in A139221.

			If k = 11 then 41 + (k + k^2) / 2 = 107 (prime).

Cf. A000217, A056561, A139219, A139221.

[k:k in [0..530]| IsPrime(41+(k+k^2) div 2)]; // Marius A. Burtea, Feb 12 2020

Select[Table[Range[0,1000]],PrimeQ[41+(#+#^2)/2]&]

is(n)=isprime(n*(n+1)/2+41) \\ Charles R Greathouse IV, Aug 16 2015

A141489 Numbers k such that k^2 + k + 257 is prime.

Original entry on oeis.org

0, 2, 3, 4, 7, 9, 10, 11, 13, 14, 17, 18, 20, 21, 23, 24, 25, 27, 28, 30, 31, 34, 37, 41, 42, 44, 48, 49, 51, 53, 56, 59, 60, 63, 65, 66, 67, 69, 70, 73, 74, 77, 79, 80, 81, 83, 88, 90, 91, 93, 94, 95, 100, 101, 104, 107, 111, 114, 115, 116, 119, 122, 125, 129, 135, 137

Offset: 1

Parthasarathy Nambi, Aug 09 2008

nonn

			If k=0, then k^2 + k + 257 = 257 (prime).
If k=100, then k^2 + k + 257 = 10357 (prime).

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

Cf. A002837, A056561, A133157, A133160.

[n: n in [0..5000] |IsPrime(n^2+n+257)]; // Vincenzo Librandi, Nov 25 2010

Select[Range[0,200],PrimeQ[#^2+#+257]&] (* Harvey P. Dale, Jun 07 2016 *)

isok(n) = isprime(n^2+n+257); \\ Michel Marcus, Mar 12 2017

More terms from Vincenzo Librandi, Mar 25 2010

A228123 Number of primes generated from Euler's polynomial x^2 + x + 41 from x = 1 to 10^n.

Original entry on oeis.org

1, 10, 86, 581, 4148, 31984, 261080, 2208196, 19132652, 168806740, 1510676802

Offset: 0

Shyam Sunder Gupta, Aug 11 2013

			a(4) = 4148 because the number of primes generated from Euler's polynomial x^2 + x + 41 from x = 1 to 10^4 are 4148.

Cf. A005846, A056561, A331876.

a = 0; n = 1; t = {}; Do[If[PrimeQ[x^2 + x + 41], a = a + 1]; If[Mod[x, n] == 0, n = n*10; AppendTo[t, a]], {x, 1, 1000000000}]; t

a(n) = A331876(n) - 1. - Amiram Eldar, Sep 23 2023

a(10) from Amiram Eldar, Sep 23 2023

A250394 Numbers k such that 56211383760397 + 44546738095860*k is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 27, 40, 64, 72, 73, 74, 80, 82, 86, 90, 91, 92, 93, 94, 98, 105, 109, 114, 123, 124, 136, 137, 146, 153, 156, 158, 159, 160, 166, 183, 185, 186, 194, 199, 204, 213, 216, 217, 228

Offset: 1

Vincenzo Librandi, Nov 21 2014

Terms up to 22 are consecutive. Arithmetic progression found in 2004 by Markus Frind, Paul Underwood, and Paul Jobling (see Green and Tao, 2008).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, Annals of Mathematics, Vol. 167, No. 2 (2008), pp. 481-547; arXiv preprint, arXiv:math/0404188 [math.NT], 2004-2007.

Cf. A002837, A056561, A180688, A250395.

[n: n in [0..300] | IsPrime(56211383760397+44546738095860*n)];

Select[Range[0, 300], PrimeQ[56211383760397 + 44546738095860 #]&]

is(n)=isprime(56211383760397+44546738095860*n) \\ Charles R Greathouse IV, Jun 13 2017

A250395 Numbers k such that 11410337850553 + 4609098694200*k is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 26, 30, 31, 41, 43, 50, 57, 61, 69, 75, 88, 90, 98, 99, 101, 108, 116, 127, 128, 131, 132, 133, 146, 154, 156, 159, 160, 162, 164, 165, 171, 172, 182, 183, 188, 191, 193, 194, 197

Offset: 1

Vincenzo Librandi, Nov 21 2014

Terms up to 21 are consecutive. Arithmetic progression found by Pritchard et al. (1995).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, Annals of Mathematics, Vol. 167, No. 2 (2008), pp. 481-547; arXiv preprint, arXiv:math/0404188 [math.NT], 2004-2007.
Paul A. Pritchard, Andrew Moran and Anthony Thyssen, Twenty-two primes in arithmetic progression, Mathematics of Computation, Vol. 64, No. 211 (1995), pp. 1337-1339.

Cf. A002837, A056561, A180688, A250394.

[n: n in [0..200] | IsPrime(11410337850553+4609098694200*n)];

Select[Range[0, 300], PrimeQ[11410337850553 + 4609098694200 #] &]

is(n)=isprime(11410337850553+4609098694200*n) \\ Charles R Greathouse IV, Jun 13 2017

A215714 Sophie Germain primes q such that q^2 + q + 41 is prime.

Original entry on oeis.org

2, 3, 5, 11, 23, 29, 53, 83, 113, 131, 179, 191, 233, 281, 293, 359, 419, 509, 641, 653, 659, 683, 719, 743, 809, 911, 953, 1013, 1019, 1103, 1289, 1439, 1481, 1511, 1601, 1733, 1811, 1901, 1931, 2003, 2039, 2339, 2393, 2549, 2693, 2903, 2939, 3023, 3299, 3329

Offset: 1

Pierre CAMI, Aug 21 2012

nonn

By definition, p = 2 * q + 1 is prime and (p^2 + 163)/4 = q^2 + q + 41 is also prime.

			29 is in the sequence because not only is 2 * 29 + 1 = 59 a prime, so is 29^2 + 29 + 41 = 911.

Pierre CAMI, Table of n, a(n) for n = 1..10000

Cf. A005384, A005846.

Select[Prime[Range[500]], PrimeQ[2# + 1] && PrimeQ[#^2 + # + 41] &] (* Alonso del Arte, Aug 21 2012 *)

A005384 INTERSECT A056561. - R. J. Mathar, Aug 23 2012

A215725 Numbers x such that 2x+1 is prime as is 2x+3 and x^2+x+41 and (x+1)^2+(x+1)+41.

Original entry on oeis.org

1, 2, 5, 8, 14, 20, 29, 35, 50, 53, 68, 74, 98, 113, 119, 134, 230, 404, 413, 509, 575, 650, 713, 725, 809, 893, 935, 938, 974, 1013, 1043, 1133, 1190, 1400, 1625, 1730, 1778, 1958, 2045, 2318, 2510, 2933, 2939, 3224, 3344, 3389, 3743, 3773

Offset: 1

Pierre CAMI, Aug 22 2012

nonn

Subsequence of A056561.

p=x^2+x+41 and q = (x+1)^2+(x+1)+41, (x+p)^2+x+p+41 = p*q is a semiprime.

			1^2+1+41=43=p , 2^2+2+41=47=q , (43+1)^2+(43+1)+41=p*q.

Pierre CAMI, Table of n, a(n) for n = 1..1300

Cf. A056561, A215697.

Select[Range[4000],And@@PrimeQ[{2#+1,2#+3,#^2+#+41,(#+1)^2+#+42}]&] (* Harvey P. Dale, Jun 02 2014 *)

A253239 Numbers k such that k^2 + k + 72491 is prime.

Original entry on oeis.org

1, 2, 3, 6, 7, 11, 12, 13, 14, 16, 17, 19, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 53, 55, 56, 57, 58, 59, 64, 65, 66, 67, 72, 73, 74, 75, 77, 78, 81, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 98, 100

Offset: 1

Eric Chen, Apr 19 2015

Of the first 10000 natural numbers, 4534 are in this sequence, making the density about 45%, quite large! (However, 72491 is not prime; it equals 71*1021, so no multiples of 71 or 1021 are in this sequence.)

			k       k^2 + k + 72491
0       72491 = 71*1021
1       72493 (prime)
2       72497 (prime)
3       72503 (prime)
4       72511 = 59*1229
5       72521 = 47*1543
6       72533 (prime)
7       72547 (prime)
8       72563 = 149*487
9       72581 = 181*401
etc.

Eric Chen, Table of n, a(n) for n = 1..4534 (all terms up to 10000)
C. Rivera, Prime producing polynomials
Eric Weisstein's World of Mathematics, Prime-generating polynomial

Cf. A048097, A028823, A056561, A141489, A160548, A116206, A050267, A128878.

[n: n in [0..100] | IsPrime(n^2 + n + 72491)]; // Vincenzo Librandi, Apr 20 2015

select(t -> isprime(t^2+t+72491), [$0..100]);

Select[Range[100], PrimeQ[#^2 + # + 72491] &]

v=[ ]; for(n=0, 100, if(isprime(n^2+n+72491), v=concat(v, n), )); v

cp's OEIS Frontend

A215697 Primes p = 2x + 1 such that x^2 + x + 41 is prime.

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A319906 Number of prime numbers of the form k^2 + k + 41 below 10^n.

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A139220 Numbers k such that 41+(k+k^2)/2 = 41+A000217(k) is prime.

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Magma

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A141489 Numbers k such that k^2 + k + 257 is prime.

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A228123 Number of primes generated from Euler's polynomial x^2 + x + 41 from x = 1 to 10^n.

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A250394 Numbers k such that 56211383760397 + 44546738095860*k is prime.

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Magma

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PARI

A250395 Numbers k such that 11410337850553 + 4609098694200*k is prime.

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Magma

Mathematica

PARI

A215714 Sophie Germain primes q such that q^2 + q + 41 is prime.

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A215725 Numbers x such that 2x+1 is prime as is 2x+3 and x^2+x+41 and (x+1)^2+(x+1)+41.