A007519 -id:A007519 - cp's OEIS frontend

A023228 Numbers k such that k and 8*k + 1 are both prime.

2, 5, 11, 17, 29, 71, 101, 107, 131, 137, 149, 179, 239, 269, 347, 401, 431, 449, 479, 491, 509, 557, 599, 617, 659, 677, 761, 809, 821, 857, 929, 941, 947, 977, 1151, 1187, 1229, 1289, 1307, 1361, 1367, 1409, 1487, 1559, 1571, 1601, 1619, 1667, 1697, 1811, 1871

Offset: 1

David W. Wilson

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Samuel S. Wagstaff, Jr., Sum of Reciprocals of Germain Primes, Journal of Integer Sequences, Vol. 24, No. 2 (2021), Article 21.9.5.

Cf. A007519 (primes of form 8n+1), A005123 ((( primes == 1 mod 8 ) - 1)/8). - Klaus Brockhaus, Dec 21 2008

[ p: p in PrimesUpTo(1900) | IsPrime(8*p+1) ]; // Klaus Brockhaus, Dec 21 2008

Select[Prime[Range[2000]], PrimeQ[8# + 1]&] (* Vincenzo Librandi, Feb 02 2014 *)

list(lim)=my(v=List()); forprime(p=2,lim, if(isprime(8*p+1), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Oct 20 2021

Sum_{n>=1} 1/a(n) is in the interval (1.151956749, 1.4207187) (Wagstaff, 2021). - Amiram Eldar, Nov 04 2021

A208178 Primes of the form 256*k + 1.

Original entry on oeis.org

257, 769, 3329, 7681, 7937, 9473, 10753, 11777, 12289, 13313, 14081, 14593, 15361, 17921, 18433, 19457, 22273, 23041, 23297, 25601, 26113, 26881, 30977, 31489, 32257, 36097, 36353, 37633, 37889, 39937, 40193, 40961, 41729, 43777, 45569, 46337, 49409, 49921

Offset: 1

Bruno Berselli, Feb 25 2012

Odd primes p such that -1 is a 128th power mod p. - Eric M. Schmidt, Mar 27 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A065091, A002144, A007519, A094407, A133870, A142925, A208177, A076339.

a208178 n = a208178_list !! (n-1)
a208178_list = filter ((== 1) . a010051) [1,257..]
-- Reinhard Zumkeller, Mar 06 2012

[ p: p in PrimesUpTo(50000) | p mod 256 eq 1 ];

Mathematica
```
Select[256 Range[195] + 1, PrimeQ]
```

for(n=1,195,r=256*n+1; if(isprime(r), print1(r", ")));

a(n) ~ 128n log n. - Charles R Greathouse IV, Nov 01 2022

A343111 Numbers having exactly 2 divisors of the form 8k + 1, that is, numbers with exactly 1 divisor of the form 8k + 1 other than 1.

Original entry on oeis.org

9, 17, 18, 25, 27, 33, 34, 36, 41, 45, 49, 50, 51, 54, 57, 63, 65, 66, 68, 72, 73, 75, 82, 85, 89, 90, 97, 98, 100, 102, 105, 108, 113, 114, 117, 119, 121, 123, 125, 126, 129, 130, 132, 135, 136, 137, 144, 145, 146, 147, 150, 161, 164, 165, 169, 170, 175

Offset: 1

Jianing Song, Apr 05 2021

			63 is a term since among the divisors of 63 (namely 1, 3, 7, 9, 21 and 63), the only divisors congruent to 1 modulo 8 are 1 and 9.

Jianing Song, Table of n, a(n) for n = 1..10000

Numbers having m divisors of the form 8*k + i: A343107 (m=1, i=1), A343108 (m=0, i=3), A343109 (m=0, i=5), A343110 (m=0, i=7), this sequence (m=2, i=1), A343112 (m=1, i=3), A343113 (m=1, i=5), A141164 (m=1, i=7).
Indices of 2 in A188169.
A007519 is a subsequence.

res(n,a,b) = sumdiv(n, d, (d%a) == b)
isA343111(n) = (res(n,8,1) == 2)

A042987 Primes congruent to {2, 3, 5, 7} mod 8.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 79, 83, 101, 103, 107, 109, 127, 131, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 197, 199, 211, 223, 227, 229, 239, 251, 263, 269, 271, 277, 283, 293, 307, 311, 317, 331, 347, 349, 359, 367, 373, 379

Offset: 1

N. J. A. Sloane

Equivalently, primes p not congruent to 1 (mod 8).

In 1981 D. Weisser proved that a prime not congruent to 1 (mod 8) and >= 7 is irregular if and only if the rational number Zeta_K(-1) is p-adically integral, that is has a denominator not divisible by p, where K is the maximal real subfield of the cyclotomic field of p-th roots of unity. - From Achava Nakhash posting, see Links.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)
Achava Nakhash, Irregular Primes and Dedekind Zeta Functions

Complement in primes of A007519.

[p: p in PrimesUpTo(1200) | p mod 8 in [2, 3, 5, 7]]; // Vincenzo Librandi, Aug 08 2012

Select[Prime[Range[100]],MemberQ[{2,3,5,7},Mod[#,8]]&]  (* Harvey P. Dale, Mar 24 2011 *)

A110567 a(n) = n^(n+1) + 1.

Original entry on oeis.org

1, 2, 9, 82, 1025, 15626, 279937, 5764802, 134217729, 3486784402, 100000000001, 3138428376722, 106993205379073, 3937376385699290, 155568095557812225, 6568408355712890626, 295147905179352825857, 14063084452067724991010

Offset: 0

Jonathan Vos Post, Sep 12 2005

For n >= 2, a(n) = the n-th positive integer such that a(n) (base n) has a block of exactly n consecutive zeros.
Comments from Alexander Adamchuk, Nov 12 2006 (Start)
(2n+1)^2 divides a(2n). a(2n)/(2n+1)^2 = {1,1,41,5713,1657009,826446281,633095889817,691413758034721,...} = A081215(2n).
p divides a(p-1) for prime p. a(p-1)/p = {1,3,205,39991,9090909091,8230246567621,...} = A081209(p-1) = A076951(p-1).
p^2 divides a(p-1) for an odd prime p. a(p-1)/p^2 = {1,41,5713,826446281,633095889817,1021273028302258913,1961870762757168078553, 14199269001914612973017444081,...} = A081215(p-1).
Prime p divides a((p-3)/2) for p = {13,17,19,23,37,41,43,47,61,67,71,89, 109,113,137,139,157,163,167,181,191,...}.
Prime p divides a((p-5)/4) for p = {29,41,61,89,229,241,281,349,421,509,601,641,661,701,709,769,809,821,881,...} = A107218(n) Primes of the form 4x^2+25y^2.
Prime p divides a((p-7)/6) for p = {79,109,127,151,313,421,541,601,613,751,757,787,...}.
Prime p divides a((p-9)/8) for p = {41,337,401,521,569,577,601,857,929,937,953,977,...} A subset of A007519(n) Primes of form 8n+1.
Prime p divides a((p-11)/10) for p = {41,181,331,601,761,1021,1151,1231,1801,...}.
Prime p divides a((p-13)/12) for p = {313,337,433,1621,1873,1993,2161,2677,2833,...}. (End)

			Examples illustrating the Comment:
a(2) = 9 because the first positive integer (base 2) with a block of 2 consecutive zeros is 100 (base 2) = 4, and the 2nd is 1001 (base 2) = 9 = 1 + 2^3.
a(3) = 82 because the first positive integer (base 3) with a block of 3 consecutive zeros is 1000 (base 3) = 81, the 2nd is 2000 (base 3) = 54 and the 3rd is 10001 (base 3) = 82 = 1 + 3^4.
a(4) = 1025 because the first positive integer (base 4) with a block of 4 consecutive zeros is 10000 (base 4) = 256, the 2nd is 20000 (base 4) = 512, the 3rd is 30000 (base 4) = 768 and the 4th 100001 (base 4) = 1025 = 1 + 4^5. and the 2nd is 1001 (base 2) = 9 = 1 + 2^3.

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A007778: n^(n+1); A000312: n^n; A014566: Sierpinski numbers of the first kind: n^n + 1.

Cf also A081209, A076951, A081215. A110567.

[n^(n+1) + 1: n in [0..25]]; // G. C. Greubel, Oct 16 2017

Table[n^(n+1)+1,{n,0,30}] (* Harvey P. Dale, Oct 30 2015 *)

for(n=0,25, print1(1 + n^(n+1), ", ")) \\ G. C. Greubel, Aug 31 2017

a(n) = A007778(n) + 1.

a(n) = A110567(n) for n > 1. - Georg Fischer, Oct 20 2018

Entry revised by N. J. A. Sloane, Oct 20 2018 at the suggestion of Georg Fischer.

A140444 Primes congruent to 1 (mod 14).

Original entry on oeis.org

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

A209554 Primes that expressed in none of the forms n<+>2 and n<+>3, where the operation <+> is defined in A206853.

Original entry on oeis.org

3, 97, 193, 257, 353, 449, 577, 641, 673, 769, 929, 1153, 1217, 1249, 1409, 1601, 1697, 1889, 2017, 2081, 2113, 2273, 2593, 2657, 2689, 2753, 3041, 3137, 3169, 3329, 3361, 3457, 3617, 4001, 4129, 4289, 4481, 4513, 4673, 4801, 4993, 5153, 5281, 5441, 5569

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Mar 10 2012

How relate these to A133870? - R. J. Mathar, Mar 13 2012
If the formulated below conjecture is true, then for n>=2, A209544 and this sequence coincide with A007519 and A133870 respectively.
Let n>=3 be odd and k>=2. We say that n possesses a property S_k, if for every integer m from interval [0,n) with the Hamming distance D(m,n) in [2,k], there exists an integer h from (m,n) with D(m,h)=D(m,n).
Conjecture (A209544 and this sequence correspond to cases k=2 and k=3 respectively).
Odd n>3 possesses the property S_k iff n has the form n=2^(2*k-1)*t+1.
Example. Let k=2, t=1. Then n=9=(1001)_2. All numbers m from [0,9) with D(m,9)=2 are 0,3,5.
For m=0, we can take h=3, since 3 from (0,9) and D(0,3)=2; for m=3, we can take h=5, since 5 from (3,9) and D(3,5)=2; for m=5,  we can take h=6, since 6  from (5,9) and D(5,6)=2. - Vladimir Shevelev, Seqfan list Apr 05 2012.
For k=2 this conjecture is true (see comment in A182187). - Vladimir Shevelev, Apr 18 2012.

Cf. A209544, A182187 (n<+>2), A182209 (n<+>3).

Cf. A133870.

hammingDistance[a_,b_] := Count[IntegerDigits[BitXor[a,b],2],1]; vS[a_,b_] := NestWhile[#+1&,a,hammingDistance[a,#]=!=b&]; (* vS[a_,b_] is the least c>=a, such that the binary Hamming distance D (a,c)=b.vS[a,b] is Vladimir's a<+>b *) A209554 = Apply[Intersection, Table[Map[Prime[#]&, Complement[Range[Last[#]], #]&[Map[PrimePi[#]&, Union[Map[#[[2]]&, Cases[Map[{PrimeQ[#], #}&[vS[#,n]]&, Range[7500]], {True,_}]]]]]],{n, 2, 3}]] (* be careful with ranges near 2^x *)

A139492 Primes of the form x^2 + 5x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

7, 37, 43, 67, 79, 109, 127, 151, 163, 193, 211, 277, 331, 337, 373, 379, 421, 457, 463, 487, 499, 541, 547, 571, 613, 631, 673, 709, 739, 751, 757, 823, 877, 883, 907, 919, 967, 991, 1009, 1033, 1051, 1087, 1093, 1117, 1129, 1171, 1201, 1213, 1297, 1303

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

Reduced form is [1, 3, -3]. Discriminant = 21. Class number = 2.
Values of the quadratic form are {0, 1, 3, 4} mod 6, so this is a subsequence of A002476. - R. J. Mathar, Jul 30 2008
It can be checked that the primes p of the form x^2 + n*x*y + y^2, n >= 3, where x and y are nonnegative, depend on n mod 6 as follows: n mod 6 = 0 => p mod 12 = {1,5}; n mod 6 = 1 => p mod 12 = {1,7}; n mod 6 = 2 => p mod 12 = {1}; n mod 6 = 3 => p mod 12 = {1,5,7,11}; n mod 6 = 4 => p mod 12 = {1}; n mod 6 = 5 => p mod 12 = {1,7}. - Walter Kehowski, Jun 01 2008

			a(1) = 7 because we can write 7 = 1^2 + 5*1*1 + 1^2.

Z. I. Borevich and I. R. Shafarevich, Number Theory.
David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.

Peter Luschny, Binary Quadratic Forms
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Primes in A243172.
Cf. A002476, A007645, A007519, A033212, A033215, A068228, A107008, A107145, A107152, A139490, A139489, A139491, A139492, A141159.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

a = {}; w = 5; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a]

# uses[binaryQF]
# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([1, 5, 1])
print(Q.represented_positives(1303, 'prime')) # Peter Luschny, May 12 2021

A139502 Primes of the form x^2 + 22x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

241, 409, 601, 769, 1009, 1129, 1201, 1249, 1321, 1489, 1609, 1801, 2089, 2161, 2281, 2521, 2689, 3001, 3049, 3121, 3169, 3361, 3529, 3769, 3889, 4129, 4201, 4441, 4561, 4729, 4801, 4969, 5209, 5281, 5449, 5521, 5569, 5641, 5689, 5881, 6121, 6361, 6481

Offset: 1

Artur Jasinski, Apr 24 2008

Also primes of the form x^2 + 120y^2. - T. D. Noe, Apr 29 2008
Also primes of the form x^2+240y^2. See A140633. - T. D. Noe, May 19 2008
In base 12, the sequence is 181, 2X1, 421, 541, 701, 7X1, 841, 881, 921, X41, E21, 1061, 1261, 1301, 13X1, 1561, 1681, 18X1, 1921, 1981, 1X01, 1E41, 2061, 2221, 2301, 2481, 2521, 26X1, 2781, 28X1, 2941, 2X61, 3021, 3081, 31X1, 3241, 3281, 3321, 3361, 34X1, 3661, 3821, 3901, where X is 10 and E is 11. Moreover, the discriminant is 340. - Walter Kehowski, Jun 01 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139489, A007645, A068228, A007519, A033212, A033212, A107152, A107008, A033215, A107145, A139490, A139491.

Cf. A139643.

[ p: p in PrimesUpTo(7000) | p mod 120 in {1, 49}]; // Vincenzo Librandi, Jul 28 2012

QuadPrimes2[1, 0, 120, 10000] (* see A106856 *)

The primes are congruent to {1, 49} (mod 120). - T. D. Noe, Apr 29 2008

A155072 Positive integers n such that the base-2 MR-expansion of 1/n is periodic with period (n-1)/4.

Original entry on oeis.org

17, 41, 97, 137, 193, 313, 401, 409, 449, 521, 569, 761, 769, 809, 857, 929, 977, 1009, 1129, 1297, 1361, 1409, 1489, 1697, 1873, 1993, 2081, 2137, 2153, 2161, 2297, 2377, 2417, 2521, 2609, 2617, 2633, 2713, 2729, 2753, 2777, 2801, 2897, 3001

Offset: 1

John W. Layman, Jan 19 2009

nonn

See A136042 for the definition of the MR-expansion of a positive real number.
It appears that all terms of this sequence are primes of the form 8n+1 (A007519).
Apparently a subsequence of A115591. - Mia Boudreau, Jun 17 2025

			Applying the definition of the base-2 MR-expansion to 1/17 gives 1/17 -> 2/17 -> 4/17 -> 8/17 -> 16/17 -> 32/17 -> 15/17 -> 30/17 -> 13/17 -> 26/17 -> 9/17 -> 18/17 -> 1/17 -> ..., which shows that the expansion begins {5,1,1,1,...} and has period 4=(17-1)/4.

Mia Boudreau, Table of n, a(n) for n = 1..1000

Cf. A136042, A007519, A115591, A136043.

a[p_] := 1 + Sum[2 Cos[2^n Pi/((2 p + 1) )], {n, 1, p}];
Select[Range[500], Reduce[a[#]^2 == 2 # + 1, Integers] &];
2 % + 1 (* Gerry Martens, May 01 2016 *)

More terms from Mia Boudreau, Jun 17 2025

cp's OEIS Frontend

A023228 Numbers k such that k and 8*k + 1 are both prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A208178 Primes of the form 256*k + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

Formula

A343111 Numbers having exactly 2 divisors of the form 8*k + 1, that is, numbers with exactly 1 divisor of the form 8*k + 1 other than 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A042987 Primes congruent to {2, 3, 5, 7} mod 8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

A110567 a(n) = n^(n+1) + 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A140444 Primes congruent to 1 (mod 14).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Formula

Extensions

A209554 Primes that expressed in none of the forms n<+>2 and n<+>3, where the operation <+> is defined in A206853.

Views

Author

Keywords

Comments

Crossrefs

Programs

A343111 Numbers having exactly 2 divisors of the form 8k + 1, that is, numbers with exactly 1 divisor of the form 8k + 1 other than 1.