A048097 -id:A048097 - cp's OEIS frontend

A048059 Primes of the form k^2 + k + 11.

11, 13, 17, 23, 31, 41, 53, 67, 83, 101, 167, 193, 251, 283, 317, 353, 431, 563, 661, 823, 881, 941, 1201, 1493, 1571, 1733, 2081, 2267, 2663, 2767, 3203, 3433, 3671, 3793, 3917, 4567, 4703, 5413, 5711, 6173, 6491, 6653, 6983, 7151, 7321, 8753, 8941, 9323, 10111

Offset: 1

N. J. A. Sloane

From Peter Bala, Apr 15 2018: (Start)
The polynomial P(n) := n^2 + n + 11 takes distinct prime values for the 10 consecutive integers n = 0 to 9. It follows that the polynomial P(n-10) = (n - 10)^2 + (n - 10) + 11 takes prime values for the 20 consecutive integers n = 0 to 19, consisting of the 10 primes above each taken twice. We note two consequences of this fact.
1) The polynomial P(2*n-10) = 4*n^2 - 38*n + 101 also takes prime values for the 10 consecutive integers n = 0 to 9.
2)The polynomial P(3*n-10) = 9*n^2 - 57*n + 101 takes prime values for the 7 consecutive integers n = 0 to 6 (= floor(19/3)). In addition, calculation shows that P(3*n-10) also takes prime values for n from -3 to -1. Equivalently put, the polynomial P(3*n-19) = 9*n^2 - 111*n + 353 takes prime values for the 10 consecutive integers n = 0 to 9. Cf. A007635 and A005846. (End)

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

Cf. A005846, A007635, A048058, A048097, A160548.

[ a: n in [0..200] | IsPrime(a) where a is n^2+n+11 ]; // Vincenzo Librandi, Dec 07 2011

lst={}; Do[p=n^2+n+11; If[PrimeQ[p], AppendTo[lst,p]], {n,0,5*5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 27 2009 *)
Select[Table[n^2+n+11, {n,0,600}], PrimeQ] (* Vincenzo Librandi, Dec 07 2011 *)

a(n) = A048058(A048097(n)). - Elmo R. Oliveira, Apr 20 2025

A048058 a(n) = n^2 + n + 11.

Original entry on oeis.org

11, 13, 17, 23, 31, 41, 53, 67, 83, 101, 121, 143, 167, 193, 221, 251, 283, 317, 353, 391, 431, 473, 517, 563, 611, 661, 713, 767, 823, 881, 941, 1003, 1067, 1133, 1201, 1271, 1343, 1417, 1493, 1571, 1651, 1733, 1817, 1903, 1991, 2081, 2173, 2267, 2363, 2461, 2561

Offset: 0

N. J. A. Sloane

Fontebasso lists this as a prime-generating polynomial due to Legendre, but doesn't give a reference. - Charles R Greathouse IV, Jun 30 2021

Seiichi Manyama, Table of n, a(n) for n = 0..10000
P. A. Fontebasso, Un'altra formula che dà una serie limitata di numeri primi, Supplemento al Periodico di matematica (1899), p. 130.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002522, A048059, A048097, A176271.

with(combinat):seq(fibonacci(3, n)+n+10, n=0..45); # Zerinvary Lajos, Jun 07 2008

f[n_]:=n^2+n+11;f[Range[0,100]] (* Vladimir Joseph Stephan Orlovsky, Mar 12 2011*)

a(n)=n^2+n+11 \\ Charles R Greathouse IV, Jun 29 2015

For n > 4: a(n) = A176271(n+1,6). - Reinhard Zumkeller, Apr 13 2010
a(n) = 2*n + a(n-1) (with a(0)=11). - Vincenzo Librandi, Aug 06 2010
Sum_{n>=0} 1/a(n) = Pi*tanh(Pi*sqrt(43)/2)/sqrt(43). - Amiram Eldar, Jan 17 2021
From Elmo R. Oliveira, Oct 28 2024: (Start)
G.f.: (11 - 20*x + 11*x^2)/(1 - x)^3.
E.g.f.: (11 + 2*x + x^2)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A173752 a(n) = m-k where (prime(k), prime(m)) is the n-th prime pair (x^2-x+11, x^2+x+11), integer x >= 0.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 3, 4, 3, 5, 7, 5, 5, 9, 8, 10, 17, 15, 15, 16, 15, 17, 18, 20, 23, 27, 25, 30, 27, 26, 30, 32, 39, 43, 49, 48, 55, 54, 48, 64, 66, 62, 61, 62, 68, 63, 65, 77, 65, 73, 79, 85, 73, 86, 93, 98, 84, 100, 107, 113, 110, 105, 107, 121, 119, 120, 119, 121, 125, 114

Offset: 1

Juri-Stepan Gerasimov, Feb 23 2010

nonn

prime(j) is used here to refer to the j-th prime.

			The first prime pair (x^2-x+11, x^2+x+11) is obtained for x = 0: 0^2-0+11 = 11 and 0^2+0+11 = 11; 11 is the fifth prime, hence a(1) = 5-5 = 0.
The second prime pair is obtained for x = 1: 1^2-1+11 = 11 and 1^2+1+11 = 13; 11 is the fifth prime and 13 is the sixth prime, hence a(2) = 6-5 = 1.
The third prime pair is obtained for x = 2: 2^2-2+11 = 13 and 2^2+2+11 = 17; 13 is the sixth prime and 17 is the seventh prime, hence a(3) = 7-6 = 1.
The eleventh prime pair is obtained for x = 13: 13^2-13+11 = 167 and 13^2+13+11 = 193; 167 is prime(39) and 193 is prime(44), hence a(11) = 44-39 = 5.

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

Cf. A048058 (n^2+n+11), A048059 (primes of the form n^2+n+11), A048097 (n such that n^2+n+11 is prime), A175154 (prime index of A048059(n)).

PrimePi:=func< n | #PrimesUpTo(n) >; [ PrimePi(p)-PrimePi(q): x in [0..850] | IsPrime(p) and IsPrime(q) where p is x^2+x+11 where q is x^2-x+11 ]; // Klaus Brockhaus, Feb 27 2010

for x from 0 to 1000 do mp := x^2+x+11 ; kp := x^2-x+11 ; if isprime(mp) and isprime(kp) then m := numtheory[pi](mp) ; k := numtheory[pi](kp) ; printf("%d,",m-k) ; end if; end do : # R. J. Mathar, Mar 01 2010

pp[n_]:=Module[{c=n^2+11},If[AllTrue[c+{n,-n},PrimeQ],PrimePi[c+n]- PrimePi[ c-n],0]]; Join[{0},Array[pp,1000]/.(0->Nothing)] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 15 2017 *)

Edited and extended by Klaus Brockhaus, Feb 27 2010

a(15) corrected and sequence extended by R. J. Mathar, Mar 01 2010

A184902 Primes that are not factors of m^2 + m + 11 (A048058).

Original entry on oeis.org

2, 3, 5, 7, 19, 29, 37, 61, 71, 73, 89, 113, 131, 137, 149, 151, 157, 163, 179, 191, 199, 211, 223, 227, 233, 241, 257, 263, 277, 313, 331, 347, 349, 373, 383, 389, 409, 419, 421, 433, 449, 457, 463, 467, 491, 499, 503, 521, 523, 571, 577

Offset: 1

Zak Seidov, May 18 2011

The discriminant of this polynomial is -43. These are the primes that are not a square (mod 43). These primes are congruent to {2, 3, 5, 7, 8, 12, 18, 19, 20, 22, 26, 27, 28, 29, 30, 32, 33, 34, 37, 39, 42} (mod 43). - T. D. Noe, May 22 2011

Inert rational primes in the field Q(sqrt(-43)). - N. J. A. Sloane, Dec 25 2017

Cf. A048058 (n^2 + n + 11), A048059 (primes of the form n^2 + n + 11), A048097 (n^2 + n + 11 is prime).

Reap[Do[p = Prime[n]; ta = Table[Mod[m(m + 1) + 11, p],{m, 0, p/2 + 1}]; If[FreeQ[ta, 0], Sow[p]], {n, 1000}]][[2, 1]]
Select[Prime[Range[100]], JacobiSymbol[#, 43] == -1 &] (* T. D. Noe, May 22 2011 *)

a(n) ~ 2n log n. - Charles R Greathouse IV, May 22 2011

A175154 a(n) = prime index of A048059(n).

Original entry on oeis.org

5, 6, 7, 9, 11, 13, 16, 19, 23, 26, 39, 44, 54, 61, 66, 71, 83, 103, 121, 143, 152, 160, 197, 238, 248, 270, 313, 336, 386, 403, 453, 481, 512, 527, 542, 619, 635, 714, 752, 804, 842, 857, 898, 915, 933, 1092, 1112, 1154, 1242, 1265, 1307, 1372, 1412, 1534, 1561

Offset: 1

Klaus Brockhaus, Feb 27 2010

nonn

Indices of primes of the form n^2+n+11.
PrimePi(A048059(n)), where PrimePi(k) is the number of primes <= k (A000720).
prime(a(n)) = A048059(n).

			A048059(7) = 53 is the seventh prime of the form n^2+n+11; 53 has prime index 16 (PrimePi(53) = 16, prime(16) = 53). Hence a(7) = 16.

Cf. A048058 (n^2+n+11), A048097 (n such that n^2+n+11 is prime), A048059 (primes of the form n^2+n+11), A000720 (PrimePi(n), number of primes <= n), A173752.

PrimePi:=func< n | #PrimesUpTo(n) >; [ PrimePi(p): x in [0..120] | IsPrime(p) where p is x^2+x+11 ];

a(n) = A000720(A048059(n)).

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

A048059 Primes of the form k^2 + k + 11.

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A048058 a(n) = n^2 + n + 11.

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A173752 a(n) = m-k where (prime(k), prime(m)) is the n-th prime pair (x^2-x+11, x^2+x+11), integer x >= 0.

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A184902 Primes that are not factors of m^2 + m + 11 (A048058).

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A175154 a(n) = prime index of A048059(n).

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

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