A005846 -id:A005846 - cp's OEIS frontend

A118140 Index of A005846(n) in the primes.

13, 14, 15, 16, 18, 20, 23, 25, 30, 32, 36, 40, 45, 48, 54, 60, 65, 69, 76, 82, 89, 96, 101, 108, 116, 125, 132, 139, 147, 156, 164, 174, 184, 192, 202, 212, 220, 229, 241, 252, 283, 295, 318, 328, 342, 356, 377, 392, 407, 420, 432, 445, 472, 485, 501, 517, 531

Offset: 1

Roger L. Bagula, May 11 2006

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000720, A005846.

a = Table[PrimePi[n^2 + n + 41], {n, 0, 39}] (* correct only up to 39 *)
PrimePi[Select[Table[n^2 + n + 41, {n, 0, 100}], PrimeQ]] (* Amiram Eldar, Sep 06 2024 *)

a(n) = PrimePi(A005846(n)) = A000720(A005846(n)).

Offset corrected by Assoc. Eds. of the OEIS, Jun 15 2010

A131577 Zero followed by powers of 2 (cf. A000079).

Original entry on oeis.org

0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592

Offset: 0

Paul Curtz, Aug 29 2007, Dec 06 2007

A000079 is the main entry for this sequence.
Binomial transform of A000035.
Essentially the same as A034008 and A000079.
a(n) = a(n-1)-th even natural numbers (A005846) for n > 1. - Jaroslav Krizek, Apr 25 2009
Where record values greater than 1 occur in A083662: A000045(n)=A083662(a(n)). - Reinhard Zumkeller, Sep 26 2009
Number of compositions of natural number n into parts >0.
The signed sequence 0, 1, -2, 4, -8, 16, -32, 64, -128, 256, -512, 1024, ... is the Lucas U(-2,0) sequence. - R. J. Mathar, Jan 08 2013
In computer programming, these are the only unsigned numbers such that k&(k-1)=0, where & is the bitwise AND operator and numbers are expressed in binary. - Stanislav Sykora, Nov 29 2013
Also the 0-additive sequence: a(n) is the smallest number larger than a(n-1) which is not the sum of any subset of earlier terms, with initial values {0, 1, 2}. - Robert G. Wilson v, Jul 12 2014
Also the smallest nonnegative superincreasing sequence: each term is larger than the sum of all preceding terms. Indeed, an equivalent definition is a(0)=0, a(n+1)=1+sum_{k=0..n} a(k). - M. F. Hasler, Jan 13 2015

Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education Journal, Vol. 31, No. 1, pp. 24-28, Winter 1997.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Adi Dani, Compositions of natural numbers over arithmetic progressions
Jimmy Devillet, Bisymmetric and quasitrivial operations: characterizations and enumerations, arXiv:1712.07856 [math.RA], 2017.
J. T. Rowell, Solution Sequences for the Keyboard Problem and its Generalizations, Journal of Integer Sequences, 18 (2015), #15.10.7.
Wikipedia, Lucas sequence
Index entries for linear recurrences with constant coefficients, signature (2).
Index entries for Lucas sequences

Cf. A000079, A003945, A042950, A020406, A046045, A011782.

int is (unsigned long n) { return !(n & (n-1)); } /* Charles R Greathouse IV, Sep 15 2012 */

a131577 = (`div` 2) . a000079
a131577_list = 0 : a000079_list  -- Reinhard Zumkeller, Dec 09 2012

[(2^n-0^n)/2: n in [0..50]]; // Vincenzo Librandi, Aug 10 2011

A131577 := proc(n)
    if n =0 then
        0;
    else
        2^(n-1) ;
    end if;
end proc: # R. J. Mathar, Jul 22 2012

Floor[2^Range[-1, 33]] (* Robert G. Wilson v, Sep 02 2007 *)
Join[{0}, 2^Range[0, 60]] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)

a(n)=1<Charles R Greathouse IV, Sep 15 2012

def A131577(n): return 1<Chai Wah Wu, Sep 09 2023

a(n) = floor(2^(n-1)). - Robert G. Wilson v, Sep 02 2007
G.f.: x/(1-2*x); a(n) = (2^n-0^n)/2. - Paul Barry, Jan 05 2009
E.g.f.: exp(x)*sinh(x). - Geoffrey Critzer, Oct 28 2012
E.g.f.: x/T(0) where T(k) = 4*k+1 - x/(1 + x/(4*k+3 - x/(1 + x/T(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Mar 17 2013
a(n) = Sum_{k=0..floor((n+1)/2)} binomial(n, 2*k-1). - Taras Goy, Jan 02 2025

More terms from Robert G. Wilson v, Sep 02 2007
Edited by N. J. A. Sloane, Sep 13 2007
Edited by M. F. Hasler, Jan 13 2015

A007635 Primes of form n^2 + n + 17.

Original entry on oeis.org

17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257, 359, 397, 479, 523, 569, 617, 719, 773, 829, 887, 947, 1009, 1277, 1423, 1499, 1657, 1823, 1997, 2087, 2179, 2273, 2467, 2879, 3209, 3323, 3557, 3677, 3923, 4049, 4177, 4987, 5273

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

a(n) = A117530(7,n) for n <= 7: a(1) = A117530(7,1) = A014556(5) = 17, A117531(7) = 7. - Reinhard Zumkeller, Mar 26 2006
Note that the gaps between terms increases by 2*k from k = 1 to 15: 19 - 17 = 2, 23 - 19 = 4, 29 - 23 = 6 and so on until 257 - 227 = 30 then fails at 289 - 257 = 32 since 289 = 17^2. - J. M. Bergot, Mar 18 2017
From Peter Bala, Apr 15 2018: (Start)
The polynomial P(n):= n^2 + n + 17 takes distinct prime values for the 16 consecutive integers n = 0 to 15. It follows that the polynomial P(n - 16) takes prime values for the 32 consecutive integers n = 0 to 31, consisting of the 16 primes above each taken twice. We note two consequences of this fact.
1) The polynomial P(2*n - 16) = 4*n^2 - 62*n + 257 also takes prime values for the 16 consecutive integers n = 0 to 15.
2)The polynomial P(3*n - 16) = 9*n^2 - 93*n + 257 takes prime values for the 11 consecutive integers n = 0 to 10 ( = floor(31/3)). In addition, calculation shows that P(3*n-16) also takes prime values for n from -5 to -1. Equivalently put, the polynomial P(3*n-31) = 9*n^2 - 183*n + 947 takes prime values for the 16 consecutive integers n = 0 to 15. Cf. A005846 and A048059. (End)
The primes in this sequence are not primes in the ring of integers of Q(sqrt(-67)). If p = n^2 + n + 17, then ((2n + 1)/2 - sqrt(-67)/2)((2n + 1)/2 + sqrt(-67)/2) = p. For example, 3^2 + 3 + 17 = 29 and (7/2 - sqrt(-67)/2)(7/2 + sqrt(-67)/2) = 29 also. - Alonso del Arte, Nov 27 2019

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 115.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 96.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime-generating Polynomial.

Cf. A005846, A028823, A048059, A160548.

[a: n in [0..250]|IsPrime(a) where a is n^2+n+17]; // Vincenzo Librandi, Dec 23 2010

Select[Table[n^2 + n + 17, {n, 0, 99}], PrimeQ] (* Alonso del Arte, Nov 27 2019 *)

select(isprime, vector(100,n,n^2+n+17)) \\ Charles R Greathouse IV, Jul 12 2016

from sympy import isprime
it = (n**2 + n + 17 for n in range(250))
print([p for p in it if isprime(p)]) # Indranil Ghosh, Mar 18 2017

a(n) = A028823(n)^2 + A028823(n) + 17. - Seiichi Manyama, Mar 19 2017

A007641 Primes of the form 2*k^2 + 29.

Original entry on oeis.org

29, 31, 37, 47, 61, 79, 101, 127, 157, 191, 229, 271, 317, 367, 421, 479, 541, 607, 677, 751, 829, 911, 997, 1087, 1181, 1279, 1381, 1487, 1597, 1951, 2207, 2341, 2621, 2767, 2917, 3229, 3391, 3557, 3727, 4079, 4261, 4447, 4637, 4831, 5231, 5437

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

The first 29 terms of 2*k^2 + 29 (k = 0 to 28) are primes. This was discovered by Adrien-Marie Legendre. The sequence and its first 8 terms appear in the novel Code to Zero by Ken Follett. - Amiram Eldar, Apr 08 2017

Let P(k) = 2*k^2 + 29. The polynomial P(2*k - 28) = 8*k^2 - 224*k + 1597 produces prime values (not distinct) for k = 0 to 28. The polynomial P(3*k - 55) = 18*k^2 - 660*k + 6079 produces distinct prime values for k = 0 to 27. Cf. A050265. - Peter Bala, Apr 16 2018

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Ken Follett, Code to Zero, New York: Signet, 2001, p. 18.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Adrien-Marie Legendre, Essai sur la théorie des nombres, Paris: Duprat, 1798, p. 10.
Jitender Singh, Prime numbers and factorization of polynomials, arXiv:2411.18366 [math.NT], 2024.
Eric Weisstein's World of Mathematics, Prime-generating Polynomial.

Cf. A005846, A050265, A352800.

[a: n in [0..60] | IsPrime(a) where a is 2*n^2+29]; // Vincenzo Librandi, Mar 20 2013

Select[Table[2 n^2 + 29, {n, 0, 70}], PrimeQ] (* Vincenzo Librandi, Mar 20 2013 *)

list(lim)=my(v=List(),t); for(n=0,sqrtint((lim-29)\2), if(isprime(t=2*n^2+29), listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Jan 20 2022

Edited by Erich Friedman, Feb 09 2002

A050265 Primes of the form 2*n^2 + 11.

Original entry on oeis.org

11, 13, 19, 29, 43, 61, 83, 109, 139, 173, 211, 349, 461, 523, 659, 733, 811, 1069, 1163, 1579, 1693, 1811, 1933, 2749, 3373, 3539, 3709, 4243, 4813, 5011, 5419, 5843, 7211, 7699, 7949, 8461, 9533, 9811, 10093, 11261, 13789, 14461, 15149, 16573

Offset: 1

Eric W. Weisstein

The polynomial 2*n^2 + 11 first fails to produce a prime for n = 11, giving 253 = 11 * 23. - Alonso del Arte, Sep 04 2016

Let P(n) = 2*n^2 + 11. The polynomial P(2*n - 10) = 8*n^2 - 80*n + 11 produces prime values (not distinct) for n = 0 to 10. The polynomial P(3*n - 19) = 18*n^2 - 228*n + 733 produces distinct prime values for n = 0 to 9. - Peter Bala, Apr 16 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime-generating Polynomial.

Cf. A005846, A007641.

[a: n in [0..100] | IsPrime(a) where a is 2*n^2+11]; // Vincenzo Librandi, Dec 08 2011

Select[2Range[0, 100]^2 + 11, PrimeQ] (* Harvey P. Dale, May 20 2011 *)

a(n) = 11 + 2*(A092968(n))^2. - R. J. Mathar, Jan 03 2009

11 added by Vincenzo Librandi, Dec 08 2011

A221712 Hardy-Littlewood constant for x^2+x+41.

Original entry on oeis.org

3, 3, 1, 9, 7, 7, 3, 1, 7, 7, 4, 7, 1, 4, 2, 1, 6, 6, 5, 3, 2, 3, 5, 5, 6, 8, 5, 7, 6, 4, 9, 8, 8, 7, 9, 6, 6, 4, 6, 8, 5, 5, 4, 5, 8, 5, 6, 5, 2, 9, 8, 5, 8, 4, 9, 1, 5, 3, 9, 4, 0, 7, 2, 7, 9, 5, 0, 2, 6, 3, 3, 1, 0, 4, 2, 6, 1, 1, 8, 1, 4, 9, 7, 3, 7, 5, 5

Offset: 1

N. J. A. Sloane, Jan 26 2013

			3.31977317747142166532355685764988796646855...

Henri Cohen, Number Theory, Vol II: Analytic and Modern Tools, Springer (Graduate Texts in Mathematics 240), 2007.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 265-266.

Christian Axler and Mehdi Hassani, Carleman's inequality over prime numbers, Integers (2021) Vol. 21, #A53.
Karim Belabas and Henri Cohen, Computation of the Hardy-Littlewood constant for quadratic polynomials, PARI/GP script, 2020.
Lea Beneish and Christopher Keyes, How often does a cubic hypersurface have a rational point?, arXiv:2405.06584 [math.NT], 2024. See p. 23.
David Broadhurst, Five families of rapidly convergent evaluations of zeta values, arXiv:2401.08997 [math.NT], 2024.
Henri Cohen, High-precision calculation of Hardy-Littlewood constants, (1998).
Henri Cohen, High precision computation of Hardy-Littlewood constants. [Cached pdf version, with permission]
Stéphane Vinatier and William Reginald Alcorn, Mathematics as seen by an artist: Inspiring mathematical objects, Eur. Math. Soc. Mag. (2025) Vol. 135, 20-31. See p. 26.

Cf. A005846, A056561, A202018, A221713, A319906, A331876, A331877.

\\ See Belabas, Cohen link. Run as HardyLittlewood2(x^2+x+41)/2 after setting the required precision.

More terms from Hugo Pfoertner, Jan 31 2020

A048988 Primes of the form 4k^2 + 4k + 59.

Original entry on oeis.org

59, 67, 83, 107, 139, 179, 227, 283, 347, 419, 499, 587, 683, 787, 1019, 1283, 1427, 1579, 1907, 2083, 2267, 2459, 2659, 3083, 3307, 3539, 3779, 4027, 4283, 4547, 5099, 5387, 5683, 5987, 6299, 6619, 6947, 7283, 8707, 9467, 9859, 10259, 10667, 11083

Offset: 1

G. L. Honaker, Jr.

From Peter Bala, Apr 18 2018: (Start)
Let P(n) = 4*n^2 + 4*n + 59. The polynomial 1/2*P(n-1/2) = 2*n^2 + 29 has prime values for n from 0 to 28. See A007641. Also P(n-14) = 4*n^2 - 108*n + 787 is prime for the 28 consecutive values of n from 0 to 27.
The sequence of 28 values of the polynomial 4*P((n-2)/4) = n^2 + 232 for n from -1 to 26 is [233, 2^3*29, 233, 2^2*59, 241, 2^3*31, 257, 2^2*67, 281, 2^3*37, 313, 2^2*83, 353, 2^3*47, 401, 2^2*107, 457, 2^3*61, 521, 2^2*139, 593, 2^3*79, 673, 2^2*179, 761, 2^3*101, 857, 2^2*227], and consists of 7 groups of 4 numbers of the form p_1, 2^3*p_2, p_3, 2^2*p_4, where the p's are prime numbers. (End)

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.

Cf. A005846, A007641, A271980.

[ a: n in [0..250] | IsPrime(a) where a is 4*n^2 +4*n + 59]; // Vincenzo Librandi, Nov 19 2010

select(isprime, [4*k*(k+1)+59$k=0..100])[];  # Alois P. Heinz, Apr 16 2025

Select[(4 #^2 + 4 # + 59) & /@ Range[0, 100], PrimeQ] (* Robert Price, Apr 16 2025 *)

lista(nn) = for(k=0, nn, if(isprime(p=4*k^2+4*k+59), print1(p, ", "))); \\ Altug Alkan, Apr 18 2018

A050267 Primes or negative values of primes in the sequence b(n) = 47n^2 - 1701n + 10181, n >= 0.

Original entry on oeis.org

10181, 8527, 6967, 5501, 4129, 2851, 1667, 577, -419, -1321, -2129, -2843, -3463, -3989, -4421, -4759, -5003, -5153, -5209, -5171, -5039, -4813, -4493, -4079, -3571, -2969, -2273, -1483, -599, 379, 1451, 2617, 3877, 5231, 6679, 8221, 9857, 11587, 13411, 15329, 17341, 19447, 21647, 31387

Offset: 1

Eric W. Weisstein

Terms are listed in the order of their appearance in sequence b.

This is a transformed version of the polynomial P(x) = 47*x^2 + 9*x - 5209 whose absolute value gives 43 distinct primes for -24 <= x <= 18, found by G. W. Fung in 1988. - Hugo Pfoertner, Dec 13 2019

R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004 (ISBN 0-387-20860-7); see Section A17, p. 59.
Paulo Ribenboim, The Little Book of Bigger Primes, Second Edition, Springer-Verlag New York, 2004. See p. 147.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
G. W. Fung and H. C. Williams, Quadratic polynomials which have a high density of prime values, Math. Comput. 55(191) (1990), 345-353.
Carlos Rivera, Problem 12: Prime producing polynomials, The Prime Puzzles & Problems Connection.
Jitender Singh, Prime numbers and factorization of polynomials, arXiv:2411.18366 [math.NT], 2024.
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.

Cf. A002383, A005471, A005846, A007635, A022464, A027753, A027755, A027758, A048059, A050267, A050268, A116206, A117081, A267252.

lst={};Do[p=47*n^2-1701*n+10181;If[PrimeQ[p],AppendTo[lst,p]],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 29 2009 *)
Select[Table[47n^2-1701n+10181,{n,0,50}],PrimeQ] (* Harvey P. Dale, Oct 03 2011 *)

[n | n <- apply(m->47*m^2-1701*m+10181, [0..100]), isprime(abs(n))] \\ Charles R Greathouse IV, Jun 18 2017

Edited by N. J. A. Sloane, May 10 2007
Further edited by Klaus Brockhaus, Mar 20 2010
More terms (to distinguish from quadratic) from Charles R Greathouse IV, Jun 18 2017

A271980 Numbers k such that 3k^2 + 39k + 37 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, 19, 20, 21, 22, 23, 25, 26, 29, 30, 31, 32, 33, 34, 35, 38, 39, 40, 41, 42, 44, 45, 46, 48, 49, 51, 52, 53, 54, 55, 57, 58, 59, 60, 63, 64, 66, 68, 69, 70, 71, 72, 79, 84, 86, 88, 89, 90, 91, 92

Offset: 1

Robert Price, Apr 17 2016

From Peter Bala, Apr 16 2018: (Start)
Let P(n) = 3*n^2 + 39*n + 37. The absolute values of the polynomial P(2*n - 29) = 12*n^2 - 270*n + 1429 for n from 0 to 27 are distinct primes, except at n = 14 when the value is 1.
The absolute values of the polynomial 3*P((n - 20)/3) = n^2 - n - 269 for n from 0 to 42 are either prime or 3 times a prime.
The absolute values of the polynomial 3*P((4*n - 89)/3) = 16*n^2 - 556*n + 4561 for n from 0 to 27 are either prime or 3 times a prime. (End)

			4 is in this sequence since 3*4^2 + 39*4 + 37 = 48+156+37 = 241 is prime.

Robert Price, Table of n, a(n) for n = 1..3510
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A256585, A050265 - A050268, A005846, A007641, A007635, A048988.

[n: n in [0..100] |IsPrime(3*n^2+39*n+37)]; // Vincenzo Librandi, Apr 19 2018

Select[Range[0, 100], PrimeQ[3*#^2 + 39*# + 37] &]

isok(n) = isprime(3*n^2 + 39*n + 37); \\ Michel Marcus, Apr 17 2016

lista(nn) = for(n=0, nn, if(ispseudoprime(3*n^2+39*n+37), print1(n, ", "))); \\ Altug Alkan, Apr 18 2016

A272074 Numbers k such that k^4 + 29*k^2 + 101 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, 21, 23, 26, 31, 32, 34, 35, 37, 43, 44, 45, 47, 49, 53, 56, 60, 61, 62, 66, 67, 68, 70, 71, 72, 74, 75, 79, 80, 81, 84, 85, 89, 90, 91, 93, 96, 99

Offset: 1

Robert Price, Apr 19 2016

			4 is in this sequence since 4^4 + 29*4^2 + 101 = 256+464+101 = 821 is prime.

Robert Price, Table of n, a(n) for n = 1..2264
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A050265 - A050268, A005846, A007641, A007635, A048988, A256585.

Cf. A271980, A272075.

Select[Range[0,100],PrimeQ[#^4+29#^2+101]&] (* Harvey P. Dale, Dec 15 2020 *)

lista(nn) = for(n=0, nn, if(ispseudoprime(n^4+29*n^2+101), print1(n, ", "))); \\ Altug Alkan, Apr 19 2016

cp's OEIS Frontend

A118140 Index of A005846(n) in the primes.

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A131577 Zero followed by powers of 2 (cf. A000079).

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A007635 Primes of form n^2 + n + 17.

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A007641 Primes of the form 2*k^2 + 29.

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A050265 Primes of the form 2*n^2 + 11.

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A221712 Hardy-Littlewood constant for x^2+x+41.

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A048988 Primes of the form 4*k^2 + 4*k + 59.

A048988 Primes of the form 4k^2 + 4k + 59.

A050267 Primes or negative values of primes in the sequence b(n) = 47n^2 - 1701n + 10181, n >= 0.

A271980 Numbers k such that 3k^2 + 39k + 37 is prime.