A048059 -id:A048059 - cp's OEIS frontend

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.

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

A297786 Decimal expansion of 10980011/999^3.

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Jan 06 2018

			0.011013017023031041053067083101121...

John D. Cook, Prime generating fractions

Cf. A048058, A048059, A272066, A297785, A297787.

Join[{0},RealDigits[10980011/999^3,10,120][[1]]] (* Harvey P. Dale, Aug 20 2021 *)

Sum_{k>=0} 10^(-3*k-3)*A048058(k) = 10980011/999^3.

A161726 a(n) = n^2 - 917*n + 9479.

Original entry on oeis.org

9479, 8563, 7649, 6737, 5827, 4919, 4013, 3109, 2207, 1307, 409, -487, -1381, -2273, -3163, -4051, -4937, -5821, -6703, -7583, -8461, -9337, -10211, -11083, -11953, -12821, -13687, -14551, -15413, -16273, -17131, -17987, -18841, -19693, -20543, -21391, -22237

Offset: 0

Arkadiusz Wesolowski, Jun 17 2009

A prime-generating polynomial of the form f(x) = x^2 - b*x + c.
|a(n)| are distinct primes for 0 <= n <= 29.
The values of this polynomial are never divisible by a prime less than 37. - Arkadiusz Wesolowski, Oct 11 2011

Arkadiusz Wesolowski, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005846, A007635, A048059.

[n^2-917*n+9479 : n in [0..36]]; // Arkadiusz Wesolowski, Mar 04 2011

seq(n^2-917*n+9479, n=0..36); # Arkadiusz Wesolowski, Mar 08 2011

Table[n^2 - 917*n + 9479, {n, 0, 36}] (* Arkadiusz Wesolowski, Mar 04 2011 *)

for(n=0, 36, print1(n^2-917*n+9479, ", ")); \\ Arkadiusz Wesolowski, Mar 02 2011

G.f.: (-9479 + 19874*x - 10397*x^2)/(x-1)^3. - R. J. Mathar, Mar 08 2011
From Elmo R. Oliveira, Feb 09 2025: (Start)
E.g.f.: exp(x)*(9479 - 916*x + x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

Definition and offset changed by R. J. Mathar, Jun 18 2009

A268101 Smallest prime p such that some polynomial of the form ax^2 - bx + p generates distinct primes in absolute value for x = 1 to n, where 0 < a < p and 0 <= b < p.

Original entry on oeis.org

2, 3, 5, 5, 7, 7, 11, 11, 11, 11, 13, 13, 17, 17, 17, 17, 19, 19, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 31, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 647, 1277, 1979, 2753

Offset: 1

Arkadiusz Wesolowski, Jan 26 2016

			a(1) = 2 (a prime), x^2 + 2 gives a prime for x = 1.
a(2) = 3 (a prime), 2*x^2 + 3 gives distinct primes for x = 1 to 2.
a(4) = 5 (a prime), 2*x^2 + 5 gives distinct primes for x = 1 to 4.
a(6) = 7 (a prime), 4*x^2 + 7 gives distinct primes for x = 1 to 6.
a(10) = 11 (a prime), 2*x^2 + 11 gives distinct primes for x = 1 to 10.
a(12) = 13 (a prime), 6*x^2 + 13 gives distinct primes for x = 1 to 12.
a(16) = 17 (a prime), 6*x^2 + 17 gives distinct primes for x = 1 to 16.
a(18) = 19 (a prime), 10*x^2 + 19 gives distinct primes for x = 1 to 18.
a(22) = 23 (a prime), 3*x^2 - 3*x + 23 gives distinct primes for x = 1 to 22.
a(28) = 29 (a prime), 2*x^2 + 29 gives distinct primes for x = 1 to 28.
a(29) = 31 (a prime), 2*x^2 - 4*x + 31 gives distinct primes for x = 1 to 29.
a(40) = 41 (a prime), x^2 - x + 41 gives distinct primes for x = 1 to 40.
a(41) = 647 (a prime), abs(36*x^2 - 594*x + 647) gives distinct primes for x = 1 to 41.
a(42) = 1277 (a prime), abs(36*x^2 - 666*x + 1277) gives distinct primes for x = 1 to 42.
a(43) = 1979 (a prime), abs(36*x^2 - 738*x + 1979) gives distinct primes for x = 1 to 43.
a(44) = 2753 (a prime), abs(36*x^2 - 810*x + 2753) gives distinct primes for x = 1 to 44.

Carlos Rivera, Problem 12
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

Cf. A027688, A027753, A027690, A027755, A048058, A048059, A007635, A007639, A007637, A007641, A202018, A005846, A117081, A050268, A268109.

A318791 Prime generating polynomial: a(n) = 9n^2 - 249n + 1763.

Original entry on oeis.org

1523, 1301, 1097, 911, 743, 593, 461, 347, 251, 173, 113, 71, 47, 41, 53, 83, 131, 197, 281, 383, 503, 641, 797, 971, 1163, 1373, 1601, 1847, 2111, 2393, 2693, 3011, 3347, 3701, 4073, 4463, 4871, 5297, 5741, 6203, 6683, 7181, 7697, 8231, 8783, 9353

Offset: 1

Arashdeep Singh, Dec 15 2018

This polynomial (9*n^2 - 249*n + 1763) generates 40 distinct primes in succession from n = 1 to 40.

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

Cf. A005846, A007635, A048059, A202018.

seq(9*n^2-249*n+1763,n=1..50); # Muniru A Asiru, Dec 19 2018

Array[9#^2 - 249# + 1763 &, 50] (* Amiram Eldar, Dec 15 2018 *)

From Chai Wah Wu, Feb 12 2019: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3.
G.f.: x*(-1763*x^2 + 3268*x - 1523)/(x - 1)^3. (End)
a(n) = p(41 - 3*n), where p(n) = n^2 + n + 41 is Euler's prime generating polynomial - see A202018 and A005846. - Peter Bala, Jun 10 2021
E.g.f.: exp(x)*(9*x^2 - 240*x + 1763) - 1763. - Elmo R. Oliveira, Feb 10 2025

cp's OEIS Frontend

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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A297786 Decimal expansion of 10980011/999^3.

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A161726 a(n) = n^2 - 917*n + 9479.

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A268101 Smallest prime p such that some polynomial of the form a*x^2 - b*x + p generates distinct primes in absolute value for x = 1 to n, where 0 < a < p and 0 <= b < p.

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A318791 Prime generating polynomial: a(n) = 9*n^2 - 249*n + 1763.

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Maple

Mathematica

Formula

A268101 Smallest prime p such that some polynomial of the form ax^2 - bx + p generates distinct primes in absolute value for x = 1 to n, where 0 < a < p and 0 <= b < p.

A318791 Prime generating polynomial: a(n) = 9n^2 - 249n + 1763.