A002837 -id:A002837 - cp's OEIS frontend

A228184 Numbers k such that k^2 + k + 41 is semiprime.

40, 41, 44, 49, 56, 65, 76, 81, 82, 84, 87, 89, 91, 96, 102, 104, 109, 117, 121, 122, 123, 126, 127, 130, 136, 138, 140, 143, 147, 155, 159, 161, 162, 163, 164, 170, 172, 173, 178, 184, 185, 186, 187, 190, 201, 204, 205, 207, 208, 209, 213, 215, 216, 217, 218

Offset: 1

Shyam Sunder Gupta, Aug 15 2013

Subsequence of A007634. Numbers in A007634 but not in here are 420, 431, 491, 492, 514, 533, 573, etc. (A097822). - R. J. Mathar, Aug 17 2013

			a(3) = 44 is in the sequence because 44^2 + 44 + 41 = 43*47 is semiprime.

Shyam Sunder Gupta, Table of n, a(n) for n = 1..4760

Cf. A002837, A001358, A007634.

a = {}; Do[
If[PrimeOmega[x^2 + x + 41] == 2, AppendTo[a, x]], {x, 1, 500}]; a

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).

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

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

A260678 Numbers n>0 for which n+(17-n)^2 is not prime.

Original entry on oeis.org

33, 34, 37, 42, 49, 50, 51, 53, 56, 58, 60, 65, 67, 68, 69, 71, 72, 75, 78, 82, 83, 84, 85, 86, 88, 91, 94, 95, 97, 100, 101, 102, 105, 106, 107, 110, 111, 113, 114, 116, 117, 118, 119, 122, 123, 124, 128, 129, 132, 133, 134, 135, 136, 139, 141, 143, 148, 151, 152, 153

Offset: 1

M. F. Hasler, Nov 15 2015

Motivated by the fact that n+(17-n)^2 = 1+16^2, 2+15^2, ..., 16+1^2, 17+0^2, 18+1^2, 19+2^2, ..., 32+15^2 are all prime. This has an explanation through Heegener numbers, similar to Euler's prime-generating polynomial, cf. A002837 and related crossrefs.

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

Cf. A260679 (n+(17-n)^2), A007635 (primes in that sequence = primes of the form n^2+n+17).

Cf. A002837 (n^2-n+41 is prime), A005846 (primes of form n^2+n+41), A007634 (n^2+n+41 is composite), A097823 (n^2+n+41 is not squarefree).

[n: n in [1..180] | not IsPrime(n+(17-n)^2)]; // Vincenzo Librandi, Nov 16 2015

remove(t -> isprime(t+(17-t)^2), [$1..200]); # Robert Israel, May 02 2017

Select[Range[200], !PrimeQ[# + (17 - #)^2] &] (* Vincenzo Librandi, Nov 16 2015 *)

for(n=1,999,isprime(n+(17-n)^2)||print1(n","))

A273597 min { x >= 0 | A273595(n) + prime(n)*x + x^2 is composite }, where A273595(n) is such that this is a local maximum.

Original entry on oeis.org

39, 38, 37, 35, 34, 32, 31, 29, 26, 25, 22, 20, 19, 17, 14, 12, 11, 12, 12, 12, 12, 16, 15, 12, 12, 13, 14, 13, 13, 14, 13, 13, 13, 13, 14, 14, 14, 16, 16, 16, 15, 15, 16, 16, 17

Offset: 2

M. F. Hasler, May 26 2016

nonn

See A273595 for further information and (cross)references.
From the initial values, the sequence seems strictly decreasing, with a(n+1) - a(n) = (prime(n+1) - prime(n))/2; however, this property does not persist beyond n = 16.
This is the subsequence of A273770 with indices n corresponding to odd primes 2n+1, see formula. - M. F. Hasler, Feb 17 2020

Index to entries related to primes produced by polynomials.

Cf. A002837, A007634, A005846, A097823, A144051, A187057 .. A187060, A190800, A191456 ff.

A273597(n)=A273770(prime(n)\2) \\ changed Feb 17 2020

A273597(n,p=prime(n),q=A273756(p\2))={for(x=1,oo,isprime(q+p*x+x^2)||return(x))} \\ M. F. Hasler, Feb 17 2020

a(n) = (81 - prime(n))/2 for 1 < n < 17.

a(n) = A273770((prime(n) - 1)/2). - M. F. Hasler, Feb 17 2020

Edited and extended using A273756(0..100) due to Don Reble, by M. F. Hasler, Feb 17 2020

A133157 Numbers k such that k^2 + k - 41 is prime.

Original entry on oeis.org

8, 15, 20, 21, 26, 29, 36, 45, 48, 59, 68, 69, 75, 78, 98, 99, 108, 111, 113, 120, 129, 134, 138, 140, 143, 161, 168, 185, 188, 189, 210, 213, 215, 216, 218, 224, 230, 231, 234, 251, 255, 260, 266, 273, 278, 279, 281, 290, 294, 299, 306, 308, 314, 320, 329, 356

Offset: 1

Parthasarathy Nambi, Dec 17 2007

			If k=8, then k^2 + n - 41 = 31 (prime).
If k=99, then k^2 + n - 41 = 9859 (prime).

Cf. A002837.

isA133157 := proc(n) isprime(n*(n+1)-41) ; end: for n from 1 to 400 do if isA133157(n) then printf("%d, ",n) ; fi ; od: # R. J. Mathar, Jan 08 2008

Select[Range[7, 400], PrimeQ[ #^2 + # - 41] &] (* Stefan Steinerberger, Dec 24 2007 *)

is(n)=isprime(n^2+n-41) \\ Charles R Greathouse IV, Jun 13 2017

More terms from Stefan Steinerberger, Dec 24 2007

More terms from R. J. Mathar, Jan 08 2008

A260679 a(n) = n + (17 - n)^2.

Original entry on oeis.org

257, 227, 199, 173, 149, 127, 107, 89, 73, 59, 47, 37, 29, 23, 19, 17, 17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257, 289, 323, 359, 397, 437, 479, 523, 569, 617, 667, 719, 773, 829, 887, 947, 1009, 1073, 1139, 1207, 1277, 1349, 1423, 1499, 1577, 1657

Offset: 1

M. F. Hasler, Nov 15 2015

Motivated by the fact that the first 32 terms of this sequence are primes. This has an explanation through Heegener numbers, similar to Euler's prime-generating polynomial (cf. A002837 and related crossrefs).
See also A007635 for the primes in this sequence, A260678 for indices k for which a(k) is composite.
Sequence provides all numbers m for which 4*m - 67 is a square. - Bruno Berselli, Nov 16 2015

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A007635 (primes in this sequence = primes of the form n^2 + n + 17).
Cf. A002837 (n^2 - n + 41 is prime), A005846 (primes of form n^2 + n + 41), A007634 (n^2 + n + 41 is composite), A097823 (n^2 + n + 41 is not squarefree).
Cf. A260678.

[n+(17-n)^2: n in [1..70]]; // Vincenzo Librandi, Nov 16 2015

Table[n + (17 - n)^2, {n, 70}] (* Vincenzo Librandi, Nov 16 2015 *)
LinearRecurrence[{3,-3,1},{257,227,199},60] (* Harvey P. Dale, May 12 2019 *)

PARI
```
for(n=1,99,print1(n+(17-n)^2,","))
```

G.f.: x*(257 - 544*x + 289*x^2)/(1 - x)^3.
From Elmo R. Oliveira, Feb 11 2025: (Start)
E.g.f.: exp(x)*(x^2 - 32*x + 289) - 289.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A284043 Starts of a run of at least n consecutive numbers k for which k^2 - k + 41 is composite.

Original entry on oeis.org

41, 41, 122, 162, 299, 326, 326, 1064, 1064, 1064, 1064, 1064, 5664, 5664, 5664, 5664, 9265, 9265, 9265, 22818, 22818, 37784, 37784, 47494, 100202, 100202, 100202, 167628, 167628, 167628, 167628, 167628, 167628, 167628, 167628, 176956, 176956, 176956, 1081297

Offset: 1

Amiram Eldar, Jun 14 2017

nonn

This sequence is inspired by the problem proposed by Sidney Kravitz in 1963: "It is known that f(n)=n^2-n+41 yields prime numbers for n=1, 2, ..., 40. Find a sequence of 40 consecutive values of n for which f(n) is composite." Lawrence A. Ringenberg and others suggested the solution that starts at f(1)*f(2)*...*f(40)+1 (about 4.890... * 10^101). B. A. Hausmann suggested the smaller solution that starts at f(1)*f(2)*...*f(20)-19 (about 3.213... * 10^42). The smallest solution is a(40) = 1081297.

			The values of f(n)=n^2-n+41 at 122, 123 and 124 are: 14803 = 113*131, 15047 = 41*367 and 15293 = 41*373. This is the first case of 3 consecutive composite values, thus a(3) = 122.

Thomas Koshy, Elementary Number Theory with Applications, Academic Press, 2nd edition, 2007, Chapter 2, p. 147, exercise 50.

Amiram Eldar, Table of n, a(n) for n = 1..130 (terms below 10^10)
Sidney Kravitz, Problem 527, Mathematics Magazine, Vol. 36, No. 4 (1963), p. 264.
Lawrence A. Ringenberg et al., A Prime Generator, Solutions to Problem 527, Mathematics Magazine, Vol. 37, No. 2 (1964), pp. 122-123.

Cf. A002837, A005846, A007634, A056561, A145292, A202018.

f[n_] := n^2 - n + 41; a = PrimeQ[f[Range[1, 10^7]]]; b = Split[a]; c = Length /@ b; d = Accumulate[c]; nc = Length[c]; e = {}; For[len = 0, len < 100, len++; k = 2;  While[k <= nc && c[[k]] < len, k += 2]; If[k <= nc && c[[k]] >= len, ind = d[[k - 1]] + 1; e = AppendTo[e, ind]]]; e

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A228184 Numbers k such that k^2 + k + 41 is semiprime.

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

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

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

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

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A260678 Numbers n>0 for which n+(17-n)^2 is not prime.

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A273597 min { x >= 0 | A273595(n) + prime(n)*x + x^2 is composite }, where A273595(n) is such that this is a local maximum.

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