A094407 -id:A094407 - cp's OEIS frontend

A140444 Primes congruent to 1 (mod 14).

29, 43, 71, 113, 127, 197, 211, 239, 281, 337, 379, 421, 449, 463, 491, 547, 617, 631, 659, 673, 701, 743, 757, 827, 883, 911, 953, 967, 1009, 1051, 1093, 1163, 1289, 1303, 1373, 1429, 1471, 1499, 1583, 1597, 1667, 1709, 1723, 1877, 1933, 2003, 2017, 2087

Offset: 1

Juri-Stepan Gerasimov, Jun 26 2008

From Federico Provvedi, May 24 2018: (Start)
Also primes congruent to 1 (mod 7).
For every prime p > 2, primes congruent to 1 (mod p) are also congruent to 1 (mod 2*p).
Conjecture: The monic polynomial P(x) = (x+1)^7/x - 1/x = ((x+1)^7-1)/x is irreducible but factorizable over Galois field (mod a(n)) with exactly 6 distinct irreducible factors of degree 1. Examples:
P(x) == (5 + x) (6 + x) (7 + x) (10 + x) (14 + x) (23 + x)    (mod 29)
P(x) == (3 + x) (9 + x) (23 + x) (28 + x) (33 + x) (40 + x)   (mod 43)
P(x) == (24 + x) (27 + x) (35 + x) (40 + x) (42 + x) (52 + x) (mod 71)
P(x) == (5 + x) (8 + x) (65 + x) (84 + x) (86 + x) (98 + x)   (mod 113)
... (End).
Primes in A131877. - Eric Chen, Jun 14 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 526.

Cf. A045437, A045458, A045471, A045473.
A090613 gives prime index.
Cf. A090614.
Cf. A131877.
Primes congruent to 1 (mod k): A000040 (k=1), A065091 (k=2), A002476 (k=3 and 6), A002144 (k=4), A030430 (k=5 and 10), this sequence (k=7 and 14), A007519 (k=8), A061237 (k=9 and 18), A141849 (k=11 and 22), A068228 (k=12), A268753 (k=13 and 26), A132230 (k=15 and 30), A094407 (k=16), A129484 (k=17 and 34), A141868 (k=19 and 38), A141881 (k=20), A124826 (k=21 and 42), A212374 (k=23 and 46), A107008 (k=24), A141927 (k=25 and 50), A141948 (k=27 and 54), A093359 (k=28), A141977 (k=29 and 58), A142005 (k=31 and 62), A133870 (k=32).

Filtered(Filtered([1..2300],n->n mod 14=1),IsPrime); # Muniru A Asiru, Jun 27 2018

[p: p in PrimesUpTo(3000)|p mod 14 in {1}]; // Vincenzo Librandi, Dec 18 2010

select(isprime,select(n->modp(n,14)=1,[$1..2300])); # Muniru A Asiru, Jun 27 2018

Select[Prime[Range[500]], Mod[#, 14] == 1 &]  (* Harvey P. Dale, Mar 21 2011 *)

is(n)=isprime(n) && n%14==1 \\ Charles R Greathouse IV, Jul 02 2016

a(n) ~ 6n log n. - Charles R Greathouse IV, Jul 02 2016

Simpler definition from N. J. A. Sloane, Jul 11 2008

A141194 Primes of the form 16k+7.

Original entry on oeis.org

7, 23, 71, 103, 151, 167, 199, 263, 311, 359, 439, 487, 503, 599, 631, 647, 727, 743, 823, 839, 887, 919, 967, 983, 1031, 1063, 1223, 1303, 1319, 1367, 1399, 1447, 1511, 1543, 1559, 1607, 1783, 1831, 1847, 1879, 2039, 2087, 2311, 2423, 2503, 2551, 2647

Offset: 1

T. D. Noe, Jun 12 2008

nonn

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A094407, A091968, A127589, A105126, A141195, A141196, A127576.

lst={};Do[If[PrimeQ[p=16*n+7],AppendTo[lst,p]],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 28 2009 *)
Select[16*Range[0,200]+7,PrimeQ] (* Harvey P. Dale, Nov 26 2015 *)

A141195 Primes of the form 16k+11.

Original entry on oeis.org

11, 43, 59, 107, 139, 251, 283, 331, 347, 379, 443, 491, 523, 571, 587, 619, 683, 811, 827, 859, 907, 971, 1019, 1051, 1163, 1259, 1291, 1307, 1451, 1483, 1499, 1531, 1579, 1627, 1723, 1787, 1867, 1931, 1979, 2011, 2027, 2203, 2251, 2267, 2347, 2411, 2459

Offset: 1

T. D. Noe, Jun 12 2008

nonn

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A094407, A091968, A127589, A141194, A105126, A141196, A127576.

lst={};Do[If[PrimeQ[p=16*n+11],AppendTo[lst,p]],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 28 2009 *)
Select[16*Range[0,200]+11,PrimeQ] (* Harvey P. Dale, Aug 22 2014 *)

A141196 Primes of the form 16k+13.

Original entry on oeis.org

13, 29, 61, 109, 157, 173, 269, 317, 349, 397, 461, 509, 541, 557, 653, 701, 733, 797, 829, 877, 941, 1021, 1069, 1117, 1181, 1213, 1229, 1277, 1373, 1453, 1549, 1597, 1613, 1693, 1709, 1741, 1789, 1901, 1933, 1949, 1997, 2029, 2141, 2221, 2237, 2269

Offset: 1

T. D. Noe, Jun 12 2008

nonn

Which sequence, this or A141194, produces more primes? The race is very close. For example, A141194(1000)=80599 and A141196(1000)=80909, a difference of just 32 primes after a thousand terms. - Art Baker, Aug 07 2019

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A094407, A091968, A127589, A141194, A105126, A141195, A127576.

lst={};Do[If[PrimeQ[p=16*n+13],AppendTo[lst,p]],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 28 2009 *)

A155943 Primes p such that 16*p + 1 is also prime.

Original entry on oeis.org

7, 37, 61, 97, 151, 163, 181, 193, 271, 313, 331, 337, 397, 421, 487, 523, 547, 571, 643, 691, 727, 757, 853, 877, 967, 1033, 1087, 1093, 1231, 1237, 1297, 1303, 1423, 1471, 1567, 1657, 1747, 1777, 1801, 1831, 1867, 1987, 2083, 2113, 2221, 2251, 2281, 2437

Offset: 1

Vincenzo Librandi, Jan 31 2009

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

Cf. A094407, A155942, A155941.

[p: p in PrimesUpTo(2500)| IsPrime(16*p + 1)]; // Vincenzo Librandi, Oct 30 2012

Select[Prime[Range[2500]], PrimeQ[(16*# + 1)]&] (* Vincenzo Librandi, Oct 30 2012 *)

A155941 Numbers n such that 16*n+1 is not prime.

Original entry on oeis.org

0, 2, 3, 4, 5, 8, 9, 10, 11, 13, 14, 17, 18, 19, 20, 23, 24, 26, 29, 30, 31, 32, 33, 34, 35, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 56, 57, 59, 60, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 77, 79, 80, 82, 83, 84, 86, 87, 89, 90, 91, 92, 94, 95

Offset: 1

Vincenzo Librandi, Jan 31 2009

			Distribution of a(n)>0 in the following triangular array:
*;
*,*;
*,*,3;
*,*,*,5;
2,*,*,*,*;
*,4,*,*,*,*;
*,*,*,*,*,*,14;
*,*,*,*,*,*,*,18;
*,*,*,*,13,*,*,*,*;
*,*,*,*,*,17,*,*,*,*;
*,*,10,*,*,*,*,*,*,*,33;
*,*,*,14,*,*,*,*,*,*,*,39;
5,*,*,*,*,*,*,*,32,*,*,*,*; etc.
where * marks the non-integer values of (2*h*k + k + h)/8 with h >= k >= 1. - _Vincenzo Librandi_, Jan 15 2013

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

Cf. A094407, A155942, A155943.

[n: n in [0..100] |not IsPrime(16*n+1)]; // Vincenzo Librandi, Oct 15 2012

Select[Range[0, 100], !PrimeQ[16 # + 1] &] (* Vincenzo Librandi, Oct 15 2012 *)

0 added by Arkadiusz Wesolowski, Aug 03 2011

A155942 Numbers n such that 16n+1 is a prime.

Original entry on oeis.org

1, 6, 7, 12, 15, 16, 21, 22, 25, 27, 28, 36, 37, 40, 42, 48, 55, 58, 61, 63, 72, 75, 76, 78, 81, 85, 88, 93, 97, 100, 106, 111, 117, 118, 126, 130, 132, 133, 135, 142, 151, 162, 163, 166, 168, 172, 175, 177, 181, 190, 193, 195, 196, 198, 201, 207, 208, 210, 216, 226

Offset: 1

Vincenzo Librandi, Jan 31 2009

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

Cf. A094407, A155941, A155943.

[n: n in [0..400] | IsPrime(16*n + 1)]; // Vincenzo Librandi, Feb 16 2013

Select[Range[400], PrimeQ[16 # + 1] &] (* Vincenzo Librandi, Feb 16 2013 *)

A325068 Prime numbers congruent to 1 modulo 16 representable neither by x^2 + 32y^2 nor by x^2 + 64y^2.

Original entry on oeis.org

17, 97, 193, 241, 401, 433, 449, 641, 673, 769, 929, 977, 1009, 1297, 1361, 1409, 1489, 1697, 1873, 2017, 2081, 2161, 2417, 2609, 2753, 2801, 2897, 3041, 3169, 3329, 3457, 3617, 3697, 3793, 3889, 4129, 4241, 4337, 4561, 4673, 5009, 5153, 5281, 5441, 5521, 5857

Offset: 1

Rémy Sigrist, Mar 27 2019

nonn

Kaplansky showed that prime numbers congruent to 1 modulo 16 are representable by both or neither of the quadratic forms x^2 + 32*y^2 and x^2 + 64*y^2. A325067 corresponds to those representable by both, and this sequence corresponds to those representable by neither.

			Regarding 17:
- 17 is a prime number,
- 17 = 16*1 + 1,
- 17 is representable neither by x^2 + 32*y^2 nor by x^2 + 64*y^2,
- hence 17 belongs to the 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 A325068
Wikipedia, Kaplansky's theorem on quadratic forms

Cf. A094407, A325067.

PARI
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See Links section.
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A125040 Primes of the form 16k+1 generated recursively. Initial prime is 17. General term is a(n)=Min {p is prime; p divides (2Q)^8 + 1}, where Q is the product of previous terms in the sequence.

Original entry on oeis.org

17, 47441, 5136468762577, 1217, 2413992194819190142614641, 113, 52654897, 241, 5310928841473, 673

Offset: 1

Nick Hobson, Nov 18 2006

All prime divisors of (2Q)^8 + 1 are congruent to 1 modulo 16.

			a(3) = 5136468762577 is the smallest prime divisor of (2Q)^8 + 1 = 45820731194492299767895461612240999140120699535617 = 5136468762577 * 33000748370307713 * 270317134666005456817, where Q = 17 * 47441.

G. A. Jones and J. M. Jones, Elementary Number Theory, Springer-Verlag, NY, (1998), p. 271.

Cf. A000945, A094407, A057204-A057208, A051308-A051335, A124984-A124993, A125037-A125045.

a = {17}; q = 1;
For[n = 2, n <= 3, n++,
    q = q*Last[a];
    AppendTo[a, Min[Select[FactorInteger[(2*q)^8 + 1][[All, 1]],
    Mod[#, 16] == 1 &]]];
    ];
a (* Robert Price, Jul 14 2015 *)

a(5)-a(10) from Max Alekseyev, Oct 18 2008

A282997 Primes of the form (p^2 + q^2)/2 such that |q^2 - p^2| is square, where p and q are prime.

Original entry on oeis.org

17, 97, 16561, 89041, 2579199841, 3497992081, 5645806321, 21103207681, 428888025121, 686770904161, 2726023770721, 4017427557361, 6831989588161, 6933052766641, 10138513506001, 19387278797041, 23452359542401, 35287577206801, 40057354132561, 62093498771041, 64116963608881

Offset: 1

Thomas Ordowski and Altug Alkan, Feb 26 2017

nonn

Primes of the form x^4 + y^4 such that q = x^2 + y^2 and p = |y^2 - x^2| are both primes.
Primes of the form n^4 + (n+1)^4 such that q = n^2 + (n+1)^2 and p = 2n+1 are both primes; so for n in A128780.
Primes of the form x^4 + y^4 such that |y^4 - x^4| is a semiprime.
From Robert G. Wilson v, Feb 26 2017: (Start)
{q, p, a(n) = (p^2+q^2)/2}
{5, 3, 17}
{13, 5, 97}
{181, 19, 16561}
{421, 29, 89041}
{71821, 379, 2579199841}
{83641, 409, 3497992081}
{106261, 461, 5645806321}
{205441, 641, 21103207681}
{926161, 1361, 428888025121}
{1171981, 1531, 686770904161}
(End)

			17 = (3^2 + 5^2)/2 and 5^2 - 3^2 = 4^2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (terms 1..59 from Robert G. Wilson v)

Subsequence of A002645 and of A094407.

Cf. A103739, A128780.

lst = {}; a = 2; While[a < 2501, b = Mod[a, 2] + 1; While[b < a, If[ PrimeQ[a^4 + b^4] && PrimeOmega[a^4 - b^4] == 2, AppendTo[lst, (a^4 + b^4)]]; b += 2]; a++]; lst (* Robert G. Wilson v, Feb 27 2017 *)

list(lim)=my(v=List(),t,n); while((t=n++^4+(n+1)^4)<=lim, if(isprime(t) && isprime(n^2+(n+1)^2) && isprime(2*n+1), listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Feb 26 2017

a(n) = A128780(n)^4 + (A128780(n)+1)^4.

a(n) == 1 (mod 16).

a(11) onward from Robert G. Wilson v, Feb 26 2017

cp's OEIS Frontend

A140444 Primes congruent to 1 (mod 14).

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A141194 Primes of the form 16k+7.

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A141195 Primes of the form 16k+11.

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A141196 Primes of the form 16k+13.

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A155943 Primes p such that 16*p + 1 is also prime.

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A155941 Numbers n such that 16*n+1 is not prime.

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A155942 Numbers n such that 16n+1 is a prime.

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A325068 Prime numbers congruent to 1 modulo 16 representable neither by x^2 + 32*y^2 nor by x^2 + 64*y^2.

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PARI

A325068 Prime numbers congruent to 1 modulo 16 representable neither by x^2 + 32y^2 nor by x^2 + 64y^2.