A141881 -id:A141881 - cp's OEIS frontend

A325071 Prime numbers congruent to 1 modulo 20 representable by both x^2 + 20y^2 and x^2 + 100y^2.

101, 181, 401, 461, 521, 541, 761, 941, 1021, 1061, 1361, 1601, 1621, 1721, 1741, 1861, 2081, 2441, 2621, 2801, 2861, 3001, 3121, 3301, 3461, 3581, 3821, 3881, 4001, 4021, 4201, 4441, 4561, 4621, 4861, 5021, 5081, 5101, 5261, 5281, 5441, 5741, 5861, 5981, 6221

Offset: 1

Rémy Sigrist, Mar 27 2019

nonn

Brink showed that prime numbers congruent to 1 modulo 20 are representable by both or neither of the quadratic forms x^2 + 20*y^2 and x^2 + 100*y^2. This sequence corresponds to those representable by both, and A325072 corresponds to those representable by neither.

			Regarding 1601:
- 1601 is a prime number,
- 1601 = 80*20 + 1,
- 1601 = 39^2 + 20*2^2 = 1^2 + 100*4^2,
- hence 1601 belongs to this sequence.

David Brink, Five peculiar theorems on simultaneous representation of primes by quadratic forms, Journal of Number Theory 129(2) (2009), 464-468, doi:10.1016/j.jnt.2008.04.007, MR 2473893.
Rémy Sigrist, PARI program for A325071
Wikipedia, Kaplansky's theorem on quadratic forms

See A325067 for similar results.

Cf. A141881, A325072.

PARI
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See Links section.
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A325072 Prime numbers congruent to 1 modulo 20 neither representable by x^2 + 20y^2 nor by x^2 + 100y^2.

Original entry on oeis.org

41, 61, 241, 281, 421, 601, 641, 661, 701, 821, 881, 1181, 1201, 1301, 1321, 1381, 1481, 1801, 1901, 2141, 2161, 2221, 2281, 2341, 2381, 2521, 2741, 3041, 3061, 3181, 3221, 3361, 3541, 3701, 3761, 4241, 4261, 4421, 4481, 4721, 4801, 5381, 5501, 5521, 5581

Offset: 1

Rémy Sigrist, Mar 27 2019

nonn

Brink showed that prime numbers congruent to 1 modulo 20 are representable by both or neither of the quadratic forms x^2 + 20*y^2 and x^2 + 100*y^2. A325071 corresponds to those representable by both, and this sequence corresponds to those representable by neither.

			Regarding 2221:
- 2221 is a prime number,
- 2221 = 111*20 + 1,
- 2221 is neither representable by x^2 + 20*y^2 nor by x^2 + 100*y^2,
- hence 2221 belongs to this sequence.

David Brink, Five peculiar theorems on simultaneous representation of primes by quadratic forms, Journal of Number Theory 129(2) (2009), 464-468, doi:10.1016/j.jnt.2008.04.007, MR 2473893.
Akihide Hanaki, Kenji Kobayashi, and Akihiro Munemasa, 3-Designs from PSL(2,q) with cyclic starter blocks, arXiv:2502.13331 [math.CO], 2025. See pp. 2, 9, 13.
Rémy Sigrist, PARI program for A325072
Wikipedia, Kaplansky's theorem on quadratic forms

See A325067 for similar results.

Cf. A141881, A325071.

PARI
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\\ See Links section.
```

A263770 Smallest prime q such that (prime(n)^2 + q*prime(n))/(prime(n) + 1) is an integer.

Original entry on oeis.org

7, 5, 7, 17, 13, 29, 19, 41, 73, 31, 97, 191, 43, 89, 97, 109, 61, 311, 137, 73, 149, 241, 337, 181, 197, 103, 313, 109, 331, 229, 257, 397, 139, 281, 151, 457, 317, 821, 337, 349, 181, 547, 193, 389, 199, 401, 1061, 449, 229, 461, 937, 241, 727, 757, 1033, 1321, 271, 1361, 557

Offset: 1

Juri-Stepan Gerasimov, Oct 25 2015

nonn

Least prime q such that q == 1 (mod prime(n) + 1).

Consists of the initial terms of sequences A002476, A002144, A002476, A007519, A068228, A140444, A061237, A141881, A107008, A132230, A133870, A141868, A124826, A142292, A142398, A141948, A088955, A142003, ...

Cf. A263729, A263730, A263769.

Table[q = 2; While[! IntegerQ[(Prime[n]^2 + q Prime@ n)/(Prime@ n + 1)], q = NextPrime@ q]; q, {n, 59}] (* Michael De Vlieger, Oct 26 2015 *)

a(n) = {p = prime(n); q = 2; while ((p^2 + p*q) % (p + 1), q = nextprime(q+1)); q;} \\ Michel Marcus, Oct 26 2015

5 is in this sequence because (prime(2)^2 + 5*prime(2))/(prime(2) + 1) = 6 and 5 is prime.

A141870 Primes congruent to 4 mod 19.

Original entry on oeis.org

23, 61, 137, 251, 479, 593, 631, 821, 859, 1049, 1087, 1163, 1201, 1277, 1429, 1543, 1619, 1657, 1733, 1847, 1999, 2113, 2341, 2417, 2531, 2683, 2797, 3253, 3329, 3557, 3671, 3709, 3823, 4013, 4051, 4127, 4241, 4507, 4583, 4621, 5039, 5077, 5153, 5381, 5419

Offset: 1

N. J. A. Sloane, Jul 11 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A105854, A141881, A141882, A141883.

[p: p in PrimesUpTo(8000) | p mod 19 eq 4 ]; // Vincenzo Librandi, Aug 15 2012

Select[Range[4,5000,19],PrimeQ[#]&] (* Vladimir Joseph Stephan Orlovsky, Mar 31 2011*)

is(n)=isprime(n) && n%19==4 \\ Charles R Greathouse IV, Jul 03 2016

a(n) ~ 18n log n. - Charles R Greathouse IV, Jul 03 2016

A141871 Primes congruent to 6 mod 19.

Original entry on oeis.org

101, 139, 367, 443, 557, 709, 823, 937, 1013, 1051, 1279, 1583, 1621, 1697, 1811, 2039, 2153, 2267, 2381, 2609, 2647, 2837, 3217, 3331, 3407, 3559, 3673, 3863, 4091, 4129, 4243, 4357, 4547, 4813, 4889, 5003, 5231, 5573, 5801, 5839, 5953, 6029, 6067, 6143

Offset: 1

N. J. A. Sloane, Jul 11 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A105854, A141881, A141882, A141883.

[p: p in PrimesUpTo(8000) | p mod 19 eq 6 ]; // Vincenzo Librandi, Aug 15 2012

Select[Range[6,5000,19],PrimeQ[#]&] (* Vladimir Joseph Stephan Orlovsky, Mar 31 2011*)
Select[Prime[Range[1000]],Mod[#,19]==6&] (* Harvey P. Dale, Sep 12 2020 *)

is(n)=isprime(n) && n%19==6 \\ Charles R Greathouse IV, Jul 03 2016

a(n) ~ 18n log n. - Charles R Greathouse IV, Jul 03 2016

A141872 Primes congruent to 7 mod 19.

Original entry on oeis.org

7, 83, 197, 311, 349, 463, 577, 653, 691, 881, 919, 1033, 1109, 1223, 1451, 1489, 1831, 1907, 2287, 2477, 2591, 2819, 2857, 2971, 3313, 3389, 3541, 3617, 3769, 4073, 4111, 4339, 4567, 4643, 4871, 4909, 5023, 5099, 5441, 5479, 5669, 5783, 5821, 5897, 6011

Offset: 1

N. J. A. Sloane, Jul 11 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A105854, A141881, A141882, A141883.

[p: p in PrimesUpTo(8000) | p mod 19 eq 7 ]; // Vincenzo Librandi, Aug 15 2012

Select[Range[7,5000,19],PrimeQ[#]&] (* Vladimir Joseph Stephan Orlovsky, Mar 31 2011*)
Select[Prime[Range[800]],Mod[#,19]==7&] (* Harvey P. Dale, May 14 2025 *)

is(n)=isprime(n) && n%19==7 \\ Charles R Greathouse IV, Jul 03 2016

a(n) ~ 18n log n. - Charles R Greathouse IV, Jul 03 2016

A141884 Primes congruent to 11 mod 20.

Original entry on oeis.org

11, 31, 71, 131, 151, 191, 211, 251, 271, 311, 331, 431, 491, 571, 631, 691, 751, 811, 911, 971, 991, 1031, 1051, 1091, 1151, 1171, 1231, 1291, 1451, 1471, 1511, 1531, 1571, 1811, 1831, 1871, 1931, 1951, 2011, 2111, 2131, 2251, 2311, 2351, 2371, 2411, 2531, 2551

Offset: 1

N. J. A. Sloane, Jul 11 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A105854, A141881, A141882, A141883.

[p: p in PrimesUpTo(3000) | p mod 20 eq 11 ]; // Vincenzo Librandi, Aug 16 2012

Select[Range[11,5000,20],PrimeQ[#]&] (* Vladimir Joseph Stephan Orlovsky, Mar 31 2011*)
Select[Prime[Range[400]],Mod[#,20]==11&] (* Harvey P. Dale, Aug 10 2021 *)

is(n)=isprime(n) && n%20==11 \\ Charles R Greathouse IV, Jul 03 2016

a(n) ~ 8n log n. - Charles R Greathouse IV, Jul 03 2016

A216257 a(n) = 840n^2 - 23100n + 86861.

Original entry on oeis.org

86861, 64601, 44021, 25121, 7901, -7639, -21499, -33679, -44179, -52999, -60139, -65599, -69379, -71479, -71899, -70639, -67699, -63079, -56779, -48799, -39139, -27799, -14779, -79, 16301, 34361, 54101, 75521, 98621, 123401, 149861, 178001, 207821, 239321, 272501

Offset: 0

Arkadiusz Wesolowski, Mar 15 2013

|a(n)| are distinct primes for 0 <= n <= 32.
The values of this polynomial are never divisible by a prime less than 79.
All terms are congruent to 1 (mod 20).

Arkadiusz Wesolowski, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A141881, A141887, A215145.

[ 840*n^2-23100*n+86861 : n in [0..34]];

Maple
```
seq(840*n^2-23100*n+86861, n=0..34);
```

Table[840*n^2 - 23100*n + 86861, {n, 0, 34}]

for(n=0, 34, print1(840*n^2-23100*n+86861, ", "))

G.f.: (86861 - 195982*x + 110801*x^2)/(1-x)^3.
From Elmo R. Oliveira, Feb 10 2025: (Start)
E.g.f.: exp(x)*(86861 - 22260*x + 840*x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

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A325071 Prime numbers congruent to 1 modulo 20 representable by both x^2 + 20*y^2 and x^2 + 100*y^2.

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A325072 Prime numbers congruent to 1 modulo 20 neither representable by x^2 + 20*y^2 nor by x^2 + 100*y^2.

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A263770 Smallest prime q such that (prime(n)^2 + q*prime(n))/(prime(n) + 1) is an integer.

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Mathematica

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A141870 Primes congruent to 4 mod 19.

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Magma

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A141871 Primes congruent to 6 mod 19.

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Magma

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A141872 Primes congruent to 7 mod 19.

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Magma

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A141884 Primes congruent to 11 mod 20.

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Magma

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A216257 a(n) = 840*n^2 - 23100*n + 86861.

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A325071 Prime numbers congruent to 1 modulo 20 representable by both x^2 + 20y^2 and x^2 + 100y^2.

A325072 Prime numbers congruent to 1 modulo 20 neither representable by x^2 + 20y^2 nor by x^2 + 100y^2.

A216257 a(n) = 840n^2 - 23100n + 86861.