A007520 -id:A007520 - cp's OEIS frontend

A007528 Primes of the form 6k-1.

5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587

Offset: 1

N. J. A. Sloane

For values of k see A024898.
Also primes p such that p^q - 2 is not prime where q is an odd prime. These numbers cannot be prime because the binomial p^q = (6k-1)^q expands to 6h-1 some h. Then p^q-2 = 6h-1-2 is divisible by 3 thus not prime. - Cino Hilliard, Nov 12 2008
a(n) = A211890(3,n-1) for n <= 4. - Reinhard Zumkeller, Jul 13 2012
There exists a polygonal number P_s(3) = 3s - 3 = a(n) + 1. These are the only primes p with P_s(k) = p + 1, s >= 3, k >= 3, since P_s(k) - 1 is composite for k > 3. - Ralf Steiner, May 17 2018
From Bernard Schott, Feb 14 2019: (Start)
A theorem due to Andrzej Mąkowski: every integer greater than 161 is the sum of distinct primes of the form 6k-1. Examples: 162 = 5 + 11 + 17 + 23 + 47 + 59; 163 = 17 + 23 + 29 + 41 + 53. (See Sierpiński and David Wells.)
{2,3} Union A002476 Union {this sequence} = A000040.
Except for 2 and 3, all Sophie Germain primes are of the form 6k-1.
Except for 3, all the lesser of twin primes are also of the form 6k-1.
Dirichlet's theorem on arithmetic progressions states that this sequence is infinite. (End)
For all elements of this sequence p=6*k-1, there are no (x,y) positive integers such that k=6*x*y-x+y. - Pedro Caceres, Apr 06 2019

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
A. Mąkowski, Partitions into unequal primes, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 8 (1960), 125-126.
Wacław Sierpiński, Elementary Theory of Numbers, p. 144, Warsaw, 1964.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, Revised edition, 1997, p. 127.

T. D. Noe, Table of n, a(n) for n = 1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
F. S. Carey, On some cases of the Solutions of the Congruence z^p^(n-1)=1, mod p, Proceedings of the London Mathematical Society, Volume s1-33, Issue 1, November 1900, Pages 294-312.
Amelia Carolina Sparavigna, The Pentagonal Numbers and their Link to an Integer Sequence which contains the Primes of Form 6n-1, Politecnico di Torino (Italy, 2021).
Amelia Carolina Sparavigna, Binary operations inspired by generalized entropies applied to figurate numbers, Politecnico di Torino (Italy, 2021).
Wikipedia, Dirichlet's theorem on arithmetic progressions.

Intersection of A016969 and A000040.
Cf. A003627, A010051, A117047, A132231, A214360, A057145.
Prime sequences A# (k,r) of the form k*n+r with 0 <= r <= k-1 (i.e., primes == r (mod k), or primes p with p mod k = r) and gcd(r,k)=1: A000040 (1,0), A065091 (2,1), A002476 (3,1), A003627 (3,2), A002144 (4,1), A002145 (4,3), A030430 (5,1), A045380 (5,2), A030431 (5,3), A030433 (5,4), A002476 (6,1), this sequence (6,5), A140444 (7,1), A045392 (7,2), A045437 (7,3), A045471 (7,4), A045458 (7,5), A045473 (7,6), A007519 (8,1), A007520 (8,3), A007521 (8,5), A007522 (8,7), A061237 (9,1), A061238 (9,2), A061239 (9,4), A061240 (9,5), A061241 (9,7), A061242 (9,8), A030430 (10,1), A030431 (10,3), A030432 (10,7), A030433 (10,9), A141849 (11,1), A090187 (11,2), A141850 (11,3), A141851 (11,4), A141852 (11,5), A141853 (11,6), A141854 (11,7), A141855 (11,8), A141856 (11,9), A141857 (11,10), A068228 (12,1), A040117 (12,5), A068229 (12,7), A068231 (12,11).
Cf. A034694 (smallest prime == 1 (mod n)).
Cf. A038700 (smallest prime == n-1 (mod n)).
Cf. A038026 (largest possible value of smallest prime == r (mod n)).
Cf. A001359 (lesser of twin primes), A005384 (Sophie Germain primes).
Cf. A048265, A324076.

Filtered(List([1..100],n->6*n-1),IsPrime); # Muniru A Asiru, May 19 2018

a007528 n = a007528_list !! (n-1)
a007528_list = [x | k <- [0..], let x = 6 * k + 5, a010051' x == 1]
-- Reinhard Zumkeller, Jul 13 2012

select(isprime,[seq(6*n-1,n=1..100)]); # Muniru A Asiru, May 19 2018

Select[6 Range[100]-1,PrimeQ]  (* Harvey P. Dale, Feb 14 2011 *)

forprime(p=2, 1e3, if(p%6==5, print1(p, ", "))) \\ Charles R Greathouse IV, Jul 15 2011

forprimestep(p=5,1000,6, print1(p", ")) \\ Charles R Greathouse IV, Mar 03 2025

A003627 \ {2}. - R. J. Mathar, Oct 28 2008
Conjecture: Product_{n >= 1} ((a(n) - 1) / (a(n) + 1)) * ((A002476(n) + 1) / (A002476(n) - 1)) = 3/4. - Dimitris Valianatos, Feb 11 2020
From Vaclav Kotesovec, May 02 2020: (Start)
Product_{k>=1} (1 - 1/a(k)^2) = 9*A175646/Pi^2 = 1/1.060548293.... =4/(3*A333240).
Product_{k>=1} (1 + 1/a(k)^2) = A334482.
Product_{k>=1} (1 - 1/a(k)^3) = A334480.
Product_{k>=1} (1 + 1/a(k)^3) = A334479. (End)
Legendre symbol (-3, a(n)) = -1 and (-3, A002476(n)) = +1, for n >= 1. For prime 3 one sets (-3, 3) = 0. - Wolfdieter Lang, Mar 03 2021

A244626 Composite numbers k congruent to 5 (mod 8) such that 2^((k-1)/2) mod k = k-1.

Original entry on oeis.org

3277, 29341, 49141, 80581, 88357, 104653, 196093, 314821, 458989, 489997, 800605, 838861, 873181, 1004653, 1251949, 1373653, 1509709, 1678541, 1811573, 1987021, 2269093, 2284453, 2387797, 2746477, 2909197, 3400013, 3429037, 3539101, 3605429, 4360621, 4502485, 5590621, 5599765

Offset: 1

Gary Detlefs, Jul 02 2014

nonn

This sequence contains the n mod 8 = 5 pseudoprimes to the following modified Fermat primality criterion:
Conjecture 1: if p is an odd prime congruent to {3,5} (mod 8) then 2^((p-1)/2) mod p = p-1.
This conjecture has been tested to 10^8.
This criterion produces far fewer pseudoprimes than the 2^(n-1) mod n = 1 test and thus has a higher probability of success. The number of pseudoprimes for the two tests up to 10^k are:
   10^5    5   26     19.23%
   10^6   13   78     16.66%
   10^7   40  228     17.54%
There are 40 terms < 10^7. If an additional constraint 3^(n-1) mod n = 1 and 5^(n-1) mod n = 1 is added, only 4 terms remain: (29341, 314821, 873181, 9863461).
This sequence appears to be a subset of A175865, A001262, A047713, A020230.
Number of terms below 10^k for k = 5..15: 5, 13, 40, 132, 369, 975, 2534, 6592, 17403, 45801, 122473. The corresponding numbers for 2^(n-1) mod n = 1: 26, 78, 228, 637, 1718, 4505, 11645, 29902, 76587, 197455, 513601. - Jens Kruse Andersen, Jul 13 2014
Also composite numbers 2n+1 with n even such that 2n+1 | 2^n+1. - Hilko Koning, Jan 27 2022
Conjecture 1 is true. With p = 2k+1 then 2^k mod (2k+1) == 2k. So 2k+1 | 2k-2^k. Prime numbers 2k+1 == +-3 (mod 8) are the prime numbers such that 2k+1 | 2^k+1 (Comments A007520). A reflection across the x-axis and +1 translation across the y-axis of the graph (2k-2^k) / (2k+1) gives the graph (2^k+1) / (2k+1). So the k values of both 2k+1 | 2k-2^k and 2k+1 | 2^k+1 are identical. - Hilko Koning, Feb 04 2022

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A001567, A003629, A070179.

for n from 5 to 10^7 by 8 do if 2^((n-1)/2) mod n = n-1 and not isprime(n) then print(n) fi od;

a(18) corrected by Jens Kruse Andersen, Jul 13 2014

A124984 Primes of the form 8*k + 3 generated recursively. Initial prime is 3. General term is a(n) = Min_{p is prime; p divides 2 + Q^2; p == 3 (mod 8)}, where Q is the product of previous terms in the sequence.

Original entry on oeis.org

3, 11, 1091, 1296216011, 2177870960662059587828905091, 76870667, 19, 257680660619, 73677606898727076965233531, 23842300525435506904690028531941969449780447746432390747, 35164737203

Offset: 1

Nick Hobson, Nov 18 2006

nonn

2+Q^2 always has a prime divisor congruent to 3 modulo 8.

			a(3) = 1091 is the smallest prime divisor congruent to 3 mod 8 of 2+Q^2 = 1091, where Q = 3 * 11.

D. M. Burton, Elementary Number Theory, McGraw-Hill, Sixth Edition (2007), p. 191.

Robert Price, Table of n, a(n) for n = 1..15
N. Hobson, Home page (listed in lieu of email address)

Cf. A000945, A007520, A057204-A057208, A051308-A051335, A124984-A124993, A125037-A125045.

a = {3}; q = 1;
For[n = 2, n ≤ 5, n++,
    q = q*Last[a];
    AppendTo[a, Min[Select[FactorInteger[2 + q^2][[All, 1]], Mod[#,
    8] \[Equal] 3 &]]];
    ];
a (* Robert Price, Jul 14 2015 *)

lista(nn) = my(f, q=3); print1(q); for(n=2, nn, f=factor(2+q^2)[, 1]~; for(i=1, #f, if(f[i]%8==3, print1(", ", f[i]); q*=f[i]; break))); \\ Jinyuan Wang, Aug 05 2022

a(10) from Robert Price, Jul 04 2015

a(11) from Robert Price, Jul 05 2015

A033200 Primes congruent to {1, 3} (mod 8); or, odd primes of form x^2 + 2*y^2.

Original entry on oeis.org

3, 11, 17, 19, 41, 43, 59, 67, 73, 83, 89, 97, 107, 113, 131, 137, 139, 163, 179, 193, 211, 227, 233, 241, 251, 257, 281, 283, 307, 313, 331, 337, 347, 353, 379, 401, 409, 419, 433, 443, 449, 457, 467, 491, 499

Offset: 1

N. J. A. Sloane

Rational primes that decompose in the field Q(sqrt(-2)). - N. J. A. Sloane, Dec 25 2017
Fermat knew of the relationship between a prime being congruent to 1 or 3 mod 8 and its being the sum of a square and twice a square, and claimed to have a firm proof of this fact. These numbers are not primes in Z[sqrt(-2)], as they have x - y sqrt(-2) as a divisor. - Alonso del Arte, Dec 07 2012
Terms m in A047471 with A010051(m) = 1. - Reinhard Zumkeller, Dec 29 2012
This sequence gives the primes p which satisfy norm(rho(p)) = + 1 with rho(p) := 2*cos(Pi/p) (the length ratio (smallest diagonal)/side in the regular p-gon). The norm of an algebraic number (over Q) is the product over all zeros of its minimal polynomial. Here norm(rho(p)) = (-1)^delta(p)* C(p, 0), with the degree delta(p) = A055034(p) = (p-1)/2. For the minimal polynomial C see A187360. For p == 1 (mod 8) the norm is C(p, 0) (see a comment on 4*A005123) and for p == 3 (mod 8) the norm is -C(p, 0) (see a comment on A186297). For the primes with norm(rho(p)) = -1 see A003628. - Wolfdieter Lang, Oct 24 2013
If p is a member then it has a unique representation as x^2+2y^2 [Frei, Theorem 3]. - N. J. A. Sloane, May 30 2014
Primes that are the quarter perimeter of a Heronian triangle. Such primes are unique to the Heronian triangle (see Yiu link). - Frank M Jackson, Nov 30 2014

			Since 11 is prime and 11 == 3 (mod 8), 11 is in the sequence. (Also 11 = 3^2 + 2 * 1^2 = (3 + sqrt(-2))(3 - sqrt(-2)).)
Since 17 is prime and 17 == 1 (mod 8), 17 is in the sequence.

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 7.

Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from T. D. Noe]
G. Frei, Euler's convenient numbers, Math. Intell. Vol. 7 No. 3 (1985), p. 56.
Zak Seidov, Table of n, a(n), x and y for n = 1..1000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Paul Yiu, CRUX, Problem 2331, Proposed by Paul Yiu
Paul Yiu and Jill S. Taylor, CRUX, Problem 2331, Solution pp 185-186
Index to sequences related to decomposition of primes in quadratic fields

Cf. A033203.

a033200 n = a033200_list !! (n-1)
a033200_list = filter ((== 1) . a010051) a047471_list
-- Reinhard Zumkeller, Dec 29 2012

[p: p in PrimesUpTo(600) | p mod 8 in [1, 3]]; // Vincenzo Librandi, Aug 04 2012

Rest[QuadPrimes2[1, 0, 2, 10000]] (* see A106856 *)
Select[Prime[Range[200]],MemberQ[{1,3},Mod[#,8]]&] (* Harvey P. Dale, Jun 09 2017 *)

is(n)=n%8<4 && n%2 && isprime(n) \\ Charles R Greathouse IV, Feb 09 2017

a(n) = A033203(n+1). - Zak Seidov, May 29 2014
A007519 UNION A007520. - R. J. Mathar, Jun 09 2020
L(-2, a(n)) = +1, n >= 1, with the Legendre symbol L. -Wolfdieter Lang, Jul 24 2024

A163183 Primes dividing 2^j + 1 for some odd j.

Original entry on oeis.org

3, 11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251, 281, 283, 307, 331, 347, 379, 419, 443, 467, 491, 499, 523, 547, 563, 571, 587, 617, 619, 643, 659, 683, 691, 739, 787, 811, 827, 859, 883, 907, 947, 971, 1019, 1033, 1049, 1051, 1091, 1097

Offset: 1

Christopher J. Smyth, Jul 22 2009

Also the primes p for which ord_p(-2) is odd, as (-2)^j == 1 (mod p).
All such p are = 1 or 3 mod 8, so sequence is subsequence of A033200, as (-2)^{j+1} == -2 (mod p) implies that (-2/p) = 1, p == 1 or 3 (mod 8).
Claim: Sequence contains all primes = 3 mod 8, so contains A007520 as a subsequence.
Proof: If p = 8r + 3 then 2^{4r+1} == 1 or -1 (mod p). If former, then (2^{2r+1})^2 == 2 (mod p), (2/p) = 1, only true for p == 1 or 7 (mod 8). So p | 2^{4r+1} + 1.
Also contains some primes == 1 (mod 8), given in A163184. So sequence is a union of A007520 and A163184.
Claim: For every p in sequence and every 2^k, the equation x^{2^k} == -2 (mod p) is soluble. Hence sequence is a subsequence of A033203 (k=1), A051071 (k=2), A051073 (k=3), A051077 (k=4), A051085 (k=5), A051101 (k=6), ....
Proof: Put x == (-2)^u (mod p). Then using (-2)^j == 1 (mod p), we can solve x^{2^k} == -2 (mod p) if can find u and v such that u*2^k + v*j = 1, possible as gcd(2^k, j) = 1.
From Jianing Song, Jun 22 2025: (Start)
The multiplicative order of -2 modulo a(n) is A385228(n).
Contained in primes congruent to 1 or 3 modulo 8 (primes p such that -2 is a quadratic residue modulo p, A033200), and contains primes congruent to 3 modulo 8 (A007520).
Conjecture: this sequence has density 7/24 among the primes (see A014663). (End)

			11 is in sequence as 11 | 2^5 + 1; 281 (smallest element of the sequence == 1 (mod 8)) is in the sequence as 281 | 2^35 + 1.

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

Sequence is a union of A007520 and A163184.
Subsequence of A033200. Contains A007520 as a subsequence.
Cf. A385228 (the actual multiplicative orders).
Cf. other bases: A014663 (base 2), A385220 (base 3), A385221 (base 4), A385192 (base 5), this sequence (base -2), A385223 (base -3), A385224 (base -4), A385225 (base -5).

with(numtheory):A:=3:p:=3: for c to 500 do p:=nextprime(p);if order(-2,p) mod 2=1 then A:=A,p;;fi;od:A;

Select[Prime[Range[200]], OddQ[MultiplicativeOrder[-2, #]] &] (* Paolo Xausa, Jun 30 2025 *)

lista(nn) = forprime(p=3, nn, if(znorder(Mod(-2, p))%2, print1(p, ", "))); \\ Jinyuan Wang, Mar 23 2020

A244628 Composite numbers n such that n == 3 (mod 8) and 2^((n-1)/2) == -1 (mod n).

Original entry on oeis.org

476971, 877099, 1302451, 1325843, 1397419, 1441091, 1507963, 1530787, 1907851, 2004403, 3090091, 3116107, 5256091, 5919187, 7883731, 9371251, 11081459, 11541307, 12263131, 13057787, 13338371, 15976747, 17134043, 18740971, 19404139, 20261251, 21623659, 22075579, 24214051

Offset: 1

Gary Detlefs, Jul 02 2014

nonn

This sequence contains the n mod 8 = 3 pseudoprimes to the following modified Fermat primality criterion:
Conjecture 1: if p is a prime congruent to {3,5} mod 8 then 2^((p-1)/2) mod p = p-1.
This conjecture has been tested to 10^8.
This modified primality test has far fewer pseudoprimes than the 2^(n-1) mod n = 1 test and thus has a much higher probability of success. The number of pseudoprimes up to 10^k for the two tests are:
   10^3 0     0
   10^4 0     2
   10^5 0     5
   10^6 2    14
   10^7 16   48
This sequence appears to be a subset of the composites in A175865.
The n mod 8 = 3 pseudoprimes are much rarer than the n mod 8 = 5 pseudoprimes. There are 16 terms < 10^7. If the additional constraints 3^(n-1) mod n = 1 and 5^(n-1) mod n = 1 are added, no terms remain.
Number of terms < 10^k: 0, 0, 0, 0, 0, 2, 16, 50, 132, ..., . - Robert G. Wilson v, Jul 21 2014
Number of terms < 10^k for k=5..15: 0, 2, 16, 50, 132, 341, 876, 2330, 6234, 16625, 44885. - Jens Kruse Andersen, Jul 27 2014
It appears that the terms of the sequence are also the composite numbers of A294912. - Hilko Koning, Dec 05 2019
Also composite numbers 2k+1 with k odd such that 2k+1 | 2^k+1. - Hilko Koning, Jan 27 2022
Conjecture 1 is true. With p = 2k+1 then 2^k mod (2k+1) == 2k. So 2k+1 | 2k-2^k . Prime numbers 2k+1 == +-3 (mod 8) are the prime numbers such that 2k+1 | 2^k+1 (Comments A007520). A reflection across the x-axis and +1 translation across the y-axis of the graph (2k-2^k) / (2k+1) gives the graph (2^k+1) / (2k+1). So the k values of both 2k+1 | 2k-2^k and 2k+1 | 2^k+1 are identical. - Hilko Koning, Feb 04 2022

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000 (first 132 terms from Robert G. Wilson v)

Cf. A003629, A070179, A175865.

for n from 3 to 10^8 by 8 do if Power(2,(n-1)/2) mod n =  n -1 and not isprime(n) then print(n) fi od

fQ[n_] := !PrimeQ@ n && PowerMod[2, (n - 1)/2, n] == n - 1; k = 3; lst = {}; While[k < 10^8, If[fQ@ k, AppendTo[lst, k]]; k += 8]; lst (* Robert G. Wilson v, Jul 21 2014 *)

A088190 Largest quadratic residue modulo prime(n).

Original entry on oeis.org

1, 1, 4, 4, 9, 12, 16, 17, 18, 28, 28, 36, 40, 41, 42, 52, 57, 60, 65, 64, 72, 76, 81, 88, 96, 100, 100, 105, 108, 112, 124, 129, 136, 137, 148, 148, 156, 161, 162, 172, 177, 180, 184, 192, 196, 196, 209, 220, 225, 228, 232, 232, 240, 249, 256, 258, 268, 268, 276

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu), Sep 22 2003

nonn

Denote a(n) by LQR(p_n). Observations (tested up to 20000 primes): - the sequence of largest QR modulo the primes (LQR(p_n) is 'almost' monotonic, - p_n-LQR(p_n) is either 1 or a prime value (see A088192) - if LQR(p_n)<=LQR(p_{n-1}) then p_n==7 mod 8 (when n>2) (see A088194) - if LQR(p_n)<=LQR(p_{n-1}) then p_n-LQR(p_n) is an odd prime, but never 5 (see A088195) For a similar set of sequences, related to quadratic non-residues, see A088196-A088201.
From Robert Israel, Oct 31 2024: (Start)
a(n) = prime(n)-1 if and only if n is 1 or in A080147.
a(n) = prime(n)-2 if and only if prime(n) is in A007520.
a(n) = prime(n)-3 if and only if prime(n) is in A107006. (End)

Robert Israel, Table of n, a(n) for n = 1..10000
F. Adorjan, The sequence of largest quadratic residues modulo the primes.

Cf. A007520, A080147, A088191, A088192, A088193, A088194, A088195, A088196, A088197, A088198, A088199, A088200, A088201, A107006.

lqr:= proc(p) local k;
  for k from p-1 by -1 do if numtheory:-quadres(k,p) = 1 then return k fi od:
end proc:
seq(lqr(ithprime(i)),i=1..100); # Robert Israel, Oct 31 2024

a[n_] := With[{p = Prime[n]}, SelectFirst[Range[p - 1, 1, -1], JacobiSymbol[#, p] == 1&]]; Array[a, 100] (* Jean-François Alcover, Feb 16 2018 *)

qrp(fr,to)= {/* Sequence of the largest QR modulo the primes */ local(m,p,v=[]); for(i=fr,to,m=1; p=prime(i); j=2; while((j<=(p-1)/2)&&(m

a(n) = max(r, r==j^2 mod p(n)|j=1, 2, ...(p(n)-1)/2)

A139487 Numbers k such that 8k + 7 is prime.

Original entry on oeis.org

0, 2, 3, 5, 8, 9, 12, 15, 18, 20, 23, 24, 27, 29, 32, 33, 38, 44, 45, 47, 53, 54, 57, 59, 60, 62, 74, 75, 78, 80, 89, 90, 92, 93, 102, 104, 107, 110, 113, 114, 120, 122, 123, 128, 129, 132, 135, 137, 143, 152, 153, 159, 162, 164, 165, 170, 174, 177, 179, 180, 183, 185

Offset: 1

Artur Jasinski, Apr 23 2008

For numbers k such that:
8k+1 is prime see A005123, primes see A007519;
8k+3 is prime see A005124, primes see A007520;
8k+5 is prime see A105133, primes see A007521;
8k+7 is prime see A139487, primes see A007522.
8k + 7 divides A000225(4k+3). - Jinyuan Wang, Mar 08 2019

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

Cf. A000225, A007519, A007520, A007521, A007522, A005123, A005124, A105133.

[n: n in [0..200] | IsPrime(8*n+7)]; // Vincenzo Librandi, Jun 25 2014

a = {}; Do[If[PrimeQ[8 n + 7], AppendTo[a, n]], {n, 0, 300}]; a
Select[Range[0,200],PrimeQ[8#+7]&] (* Harvey P. Dale, Oct 10 2012 *)

is(n)=isprime(8*n+7) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = (A007522(n) - 7)/8, n >= 1.

A141849 Primes congruent to 1 mod 11.

Original entry on oeis.org

23, 67, 89, 199, 331, 353, 397, 419, 463, 617, 661, 683, 727, 859, 881, 947, 991, 1013, 1123, 1277, 1321, 1409, 1453, 1607, 1783, 1871, 2003, 2069, 2113, 2179, 2267, 2311, 2333, 2377, 2399, 2531, 2663, 2707, 2729, 2861, 2927, 2971, 3037, 3169, 3191, 3257

Offset: 1

N. J. A. Sloane, Jul 11 2008

Conjecture: Also primes p such that ((x+1)^11-1)/x has 10 distinct irreducible factors of degree 1 over GF(p). - Federico Provvedi, Apr 17 2018

Primes congruent to 1 mod 22. - Chai Wah Wu, Apr 28 2025

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

Cf. A000040, A090187, A102656.
Prime sequences A# (k,r) of the form k*n+r with 0 <= r <= k-1 (i.e., primes == r (mod k), or primes p with p mod k = r) and gcd(r,k)=1: A000040 (1,0), A065091 (2,1), A002476 (3,1), A003627 (3,2), A002144 (4,1), A002145 (4,3), A030430 (5,1), A045380 (5,2), A030431 (5,3), A030433 (5,4), A002476 (6,1), A007528 (6,5), A140444 (7,1), A045392 (7,2), A045437 (7,3), A045471 (7,4), A045458 (7,5), A045473 (7,6), A007519 (8,1), A007520 (8,3), A007521 (8,5), A007522 (8,7), A061237 (9,1), A061238 (9,2), A061239 (9,4), A061240 (9,5), A061241 (9,7), A061242 (9,8), A030430 (10,1), A030431 (10,3), A030432 (10,7), A030433 (10,9), this sequence (11,1), A090187 (11,2), A141850 (11,3), A141851 (11,4), A141852 (11,5), A141853 (11,6), A141854 (11,7), A141855 (11,8), A141856 (11,9), A141857 (11,10), A068228 (12,1), A040117 (12,5), A068229 (12,7), A068231 (12,11).
Cf. A034694 (smallest prime == 1 (mod n)).
Cf. A038700 (smallest prime == n-1 (mod n)).
Cf. A038026 (largest possible value of smallest prime == r (mod n)).

Filtered([1..4000],n->n mod 11=1 and IsPrime(n)); # Muniru A Asiru, Apr 19 2018

[ p: p in PrimesUpTo(5000) | p mod 11 eq 1 ]; // Vincenzo Librandi, Apr 19 2011

a:=select(n->isprime(n) and modp(n,11)=1,[$1..4000]); # Muniru A Asiru, Apr 19 2018

Select[Range[1,10000,11],PrimeQ] (* Vladimir Joseph Stephan Orlovsky, May 18 2011 *)

is(n)=isprime(n) && n%11==1 \\ Charles R Greathouse IV, Jul 01 2016

forstep(n=2, 1e3, 2, if(isprime(p=11*n+1), print1(p, ", "))); \\ Altug Alkan, Apr 19 2018

a(n) ~ 10n log n. - Charles R Greathouse IV, Jul 02 2016

A343112 Numbers having exactly 1 divisor of the form 8*k + 3.

Original entry on oeis.org

3, 6, 9, 11, 12, 15, 18, 19, 21, 22, 24, 30, 35, 36, 38, 39, 42, 43, 44, 45, 48, 55, 59, 60, 63, 67, 69, 70, 72, 76, 77, 78, 83, 84, 86, 87, 88, 90, 91, 93, 95, 96, 107, 110, 111, 115, 117, 118, 120, 121, 126, 131, 133, 134, 138, 139, 140, 141, 143, 144

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 divisor congruent to 3 modulo 8 is 3.

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), A343111 (m=2, i=1), this sequence (m=1, i=3), A343113 (m=1, i=5), A141164 (m=1, i=7).
Indices of 1 in A188170.
A007520 is a subsequence.

res(n,a,b) = sumdiv(n, d, (d%a) == b)
isA343112(n) = (res(n,8,3) == 1)

cp's OEIS Frontend

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A088190 Largest quadratic residue modulo prime(n).

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