A007635 -id:A007635 - cp's OEIS frontend

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

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","))

A267069 Nonnegative numbers n such that abs(103n^2 - 4707n + 50383) 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, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 47, 49, 50, 51, 52, 53, 54, 57, 59, 60, 61, 63, 64, 65, 66, 67, 69, 73, 74, 76, 77, 80

Offset: 1

Robert Price, Apr 28 2016

nonn

43 is the smallest number not in this sequence.

See A267252 for more information. - Hugo Pfoertner, Dec 13 2019

			4 is in this sequence since 103*4^2 - 4707*4 + 50383  = 1648-18828+50383 = 33203 is prime.

Robert Price, Table of n, a(n) for n = 1..3885
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A050268, A050267, A005846, A007641, A007635, A048988, A050265, A050266.

Cf. A271980, A272030, A272074, A272075, A272160, A271144, A272285, A267252.

Select[Range[0, 100], PrimeQ[103#^2 - 4707# + 50383 ] &]

lista(nn) = for(n=0, nn, if(isprime(abs(103*n^2-4707*n+50383)), print1(n, ", "))); \\ Altug Alkan, Apr 28 2016, corrected by Hugo Pfoertner, Dec 13 2019

Title corrected by Hugo Pfoertner, Dec 13 2019

A272076 Numbers n such that abs(7n^2 - 371n + 4871) 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, 23, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 59, 61, 63, 65, 67, 68, 72, 73, 74, 75, 76, 77, 78

Offset: 1

Robert Price, Apr 19 2016

nonn

			4 is in this sequence since 7*4^2 - 371*4 + 4871 = 112-1484+4871 = 3499 is prime.

Robert Price, Table of n, a(n) for n = 1..3530
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A050268, A050267, A005846, A007641, A007635, A048988, A050265, A050266.

Cf. A271980, A272074, A272075, A272077.

Select[Range[0, 100], PrimeQ[7#^2 - 371# + 4871] &]

lista(nn) = for(n=0, nn, if(ispseudoprime(abs(7*n^2-371*n+4871)), print1(n, ", "))); \\ Altug Alkan, Apr 19 2016

A272813 Nonnegative numbers n such that n^2 - 79n + 1601 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, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66

Offset: 1

Robert Price, May 06 2016

nonn

80 is the smallest number not in this sequence.

See A005846 for the corresponding primes.

			4 is in this sequence since 4^2 - 79*4 + 1601 = 16-316+1601 = 1301 is prime.

Robert Price, Table of n, a(n) for n = 1..4179
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A050268, A050267, A005846, A007641, A007635, A048988, A050265, A050266.

Cf. A271980, A272030, A272074, A272075, A272160, A271144, A272285, A272401, A272438, A272444, A272410, A272555, A272710, A005846.

Select[Range[0, 100], PrimeQ[#^2 - 79# + 1601 ] &]

lista(nn) = for(n=0, nn, if(isprime(n^2-79*n+1601), print1(n, ", "))); \\ Altug Alkan, May 06 2016

A281437 Primes of the form 25n^2 + 25n + 47.

Original entry on oeis.org

47, 97, 197, 347, 547, 797, 1097, 1447, 1847, 2297, 2797, 3347, 3947, 4597, 5297, 6047, 8597, 9547, 11597, 12697, 17597, 18947, 20347, 23297, 24847, 28097, 31547, 33347, 37097, 39047, 41047, 45197, 49547, 51797, 61297, 66347, 68947, 71597, 74297, 77047, 79847

Offset: 1

Waldemar Puszkarz, Oct 05 2017

nonn

The first 16 terms correspond to n from 0 to 15, which makes 25*n^2 + 25*n + 47 a prime-generating polynomial (see the link).
This is a prime-generating polynomial of the form s*n^2 + s*n + p, where s=k^2 and p is prime with s and p containing at most two digits. Prime-generating polynomials of this kind arise for k=1,2,3,5,7. This is the case of k=5; it generates most primes in a row out of the prime k's listed, with 12 for k=3,7, and 14 for k=2. See also A005846 and A007635 (k=1), and A048988 (k=2).
All terms are of the form 10m+7, with their next-to-last digits being 4 or 9.

			197 is a term as it is a prime corresponding to n=2: 25*4 + 25*2 + 47 = 197.

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

Cf. A000040 (primes), A005846, A007635, A048988, A292578 (similar prime-generating sequences).

select(isprime, [seq(25*n^2 + 25*n + 47, n=0..200)]); # Robert Israel, Dec 12 2024

Select[Range[0,100]//25#^2+25#+47&, PrimeQ]

for(n=0, 100, isprime(p=25*n^2+25*n+47)&& print1(p ", "))

A297787 Decimal expansion of 16968017/999^3.

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Jan 06 2018

			0.017019023029037047059073089107127149173199227257289...

John D. Cook, Prime generating fractions

Cf. A007635, A272066, A297785, A297786.

RealDigits[16968017/999^3, 10, 111][[1]] (* or *)
RealDigits[ Sum[10^(-3k -3)*(k^2 +k +17), {k, 0, 37}], 10, 111][[1]] (* Robert G. Wilson v, Jan 07 2018 *)

Sum_{k>=0} 10^(-3*k-3)*(k^2+k+17) = 16968017/999^3.

A300473 Numbers k with the property that k^2 + 21k + 1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 51, 52, 53, 54, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 73, 77, 78, 79, 80, 81, 82, 84, 85, 86, 87, 89, 91, 97, 100

Offset: 1

James R. Buddenhagen, Mar 06 2018

nonn

The quadratic polynomial p(k) = k^2 + 21*k + 1 is not a prime-generating polynomial in the sense of Eric Weisstein's World of Mathematics (see link) because p(0) is not prime.

However p(k) is prime for the first 17 positive integral values of k and among polynomials of the form k^2 + j*k + 1, the present polynomial (j = 21) generates more primes than any polynomial of that form for any positive integral j < 231, at least for positive integers, k, in the range 0 < k < 10^6.

			17 is in the sequence because 17^2 + 21 * 17 + 1 = 647 is prime.
18 is not in the sequence because 18^2 + 21 * 18 + 1 = 703 = 19 * 37.

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

Cf. A292509, A050267, A050268, A005846, A007635, A007641, A048988.

select(k-> isprime(k^2+21*k+1), [$1..100])

Select[Range[100], PrimeQ[#^2 + 21# + 1] &] (* Alonso del Arte, Mar 06 2018 *)

isok(k) = isprime(k^2+21*k+1); \\ Altug Alkan, Mar 07 2018

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

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)

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.

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

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A267069 Nonnegative numbers n such that abs(103*n^2 - 4707*n + 50383) is prime.

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A272076 Numbers n such that abs(7*n^2 - 371*n + 4871) is prime.

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A272813 Nonnegative numbers n such that n^2 - 79n + 1601 is prime.

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A281437 Primes of the form 25*n^2 + 25*n + 47.

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A297787 Decimal expansion of 16968017/999^3.

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A300473 Numbers k with the property that k^2 + 21k + 1 is prime.

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A267069 Nonnegative numbers n such that abs(103n^2 - 4707n + 50383) is prime.

A272076 Numbers n such that abs(7n^2 - 371n + 4871) is prime.

A281437 Primes of the form 25n^2 + 25n + 47.

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.