A007522 -id:A007522 - cp's OEIS frontend

A254764 Fundamental positive solution x = x1(n) of the first class of the Pell equation x^2 - 2*y^2 = A007522(n), n >=1 (primes congruent to 7 mod 8).

3, 5, 7, 7, 11, 9, 11, 15, 13, 13, 17, 19, 15, 17, 19, 17, 19, 19, 23, 25, 23, 21, 25, 23, 27, 29, 29, 25, 27, 35, 31, 27, 29, 33, 29, 29, 31, 35, 31, 37, 43, 35, 33, 37, 33, 35, 33, 41, 47, 35

Offset: 1

Wolfdieter Lang, Feb 12 2015

For the corresponding term y1(n) see A254765(n).
For the positive fundamental proper (sometimes called primitive) solutions x2(n) and y2(n) of the second class of this (generalized) Pell equation see A254766(n) and A254929(n).
The present solutions of the first class are the smallest positive ones.
See the Nagell reference Theorem 111 p. 210 for the proof of the existence of solutions (the discriminant of this binary quadratic form is +8 hence it is an indefinite form with an infinitude of solutions if there exists at least one).
See the Nagell reference Theorem 110, p. 208 for the proof that there are only two classes of solutions for this Pell equation, because the equation is solvable, and the primes A007522(n) do not divide 4.
The present fundamental solutions are found according to the Nagell reference Theorem 108, p. 205, adapted to the case at hand, by scanning the following two inequalities for solutions x1(n) = 2*X1(n) + 1 and y1(n) = 2*Y1(n) + 1. The intervals for X1(n) and Y1(n) to be scanned are ceiling((sqrt(2+p(n))-1)/2) <= X1(n) <=  floor(sqrt((2*p(n))-1)/2), with p(n) = A007522(n) and 0 <= Y1(n) <= floor((sqrt(p(n)/2)-1)/2).
The general positive proper solutions for both classes are obtained by applying positive powers of the matrix M = [[3,4],[2,3]] on the fundamental column vectors (x(n),y(m))^T.
The least positive x solutions (that is the ones of the first class) for the primes +1 and -1 (mod 8) together (including also prime 2) are given in A002334.

			The first pairs [x1(n), y1(n)] of the fundamental positive solutions of this first class are (we list the prime A007522(n) as first entry):
  [7, [3, 1]], [23, [5, 1]], [31, [7, 3]], [47, [7, 1]], [71, [11, 5]], [79, [9, 1]], [103, [11, 3]], [127, [15, 7]], [151, [13, 3]], [167, [13, 1]], [191, [17, 7]], [199, [19, 9]], [223, [15, 1]], ...
a(3)^2 - 2*A254765(3)^2 = 7^2 - 2*3^2 = 31 = A007522(3).

T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964.

Cf. A007522, A254765, A254766, A254929, A254760, A254761, A254762, A254763, A002334.

a(n)^2 - 2*A254765(n)^2 = A007522(n) gives the smallest positive (proper) solution of this (generalized) Pell equation.

A254765 Fundamental positive solution y = y1(n) of the first class of the Pell equation x^2 - 2*y^2 = A007522(n), n >=1 (primes congruent to 7 mod 8).

Original entry on oeis.org

1, 1, 3, 1, 5, 1, 3, 7, 3, 1, 7, 9, 1, 5, 7, 3, 5, 1, 9, 11, 7, 1, 9, 5, 11, 13, 11, 3, 7, 17, 11, 1, 7, 13, 3, 1, 7, 13, 5, 15, 21, 11, 7, 13, 5, 9, 1, 17, 23, 1

Offset: 1

Wolfdieter Lang, Feb 12 2015

For the corresponding term x1(n) see A254764(n).
See A254764 for comments and the Nagell reference.
The least positive y solutions (that is the ones of the first class) for the primes +1 and -1 (mod 8) together (including also prime 2) are given in A002335.

			A254764(4)^2 - 2*a(4)^2 = 7^2 - 2*1^2 = 47 = A007522(4).

Cf. A007522, A254764, A254766, A254929, A254760, A254761, A254762, A254763, A002335.

A254764(n)^2 - 2*a(n)^2 = A007522(n) gives the smallest positive (proper) solution of this (generalized) Pell equation.

A186303 a(n) = ( A007522(n)+1 )/2.

Original entry on oeis.org

4, 12, 16, 24, 36, 40, 52, 64, 76, 84, 96, 100, 112, 120, 132, 136, 156, 180, 184, 192, 216, 220, 232, 240, 244, 252, 300, 304, 316, 324, 360, 364, 372, 376, 412, 420, 432, 444, 456, 460, 484, 492, 496, 516, 520, 532, 544

Offset: 1

Marco Matosic, Feb 17 2011

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007522, A186302.

(#+1)/2&/@Select[8*Range[0,300]+7,PrimeQ] (* Harvey P. Dale, Jul 14 2011 *)

is(n)=n%4==0&&isprime(2*n-1) \\ Charles R Greathouse IV, Jan 22 2013

a(n) = A186302(n)+1.

A186304 A007522(n)-2.

Original entry on oeis.org

5, 21, 29, 45, 69, 77, 101, 125, 149, 165, 189, 197, 221, 237, 261, 269, 309, 357, 365, 381, 429, 437, 461, 477, 485, 501, 597, 605, 629, 645, 717, 725, 741, 749, 821, 837, 861, 885, 909, 917, 965, 981, 989, 1029

Offset: 1

Marco Matosic, Feb 17 2011

nonn

Extensions to Fermat’s Little Theorem precisely indicate a composite or prime number. See A186293 for an introduction to A186293-A186305.
The sequence shows p-2 where p are the primes == 7 (mod 8).
(k*p+(p-2)) ^ (j*(p-1)+1) == (k*p+((p-1)/2)) ^ (j*(p-1)+(p-2)) == p-2 (mod p).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Select[Prime[Range[200]],Mod[#,8]==7&]-2 (* Harvey P. Dale, Dec 24 2021 *)

A157115 Alternate terms of A007519, A007520, A007521, A007522.

Original entry on oeis.org

17, 3, 5, 7, 41, 11, 13, 23, 73, 19, 29, 31, 89, 43, 37, 47, 97, 59, 53, 71, 113, 67, 61, 79, 137, 83, 101, 103, 193, 107, 109, 127, 233, 131, 149, 151, 241, 139, 157, 167, 257, 163, 173, 191, 281, 179, 181, 199, 313, 211, 197, 223, 337, 227, 229, 239, 353, 251, 269

Offset: 1

Zak Seidov and N. J. A. Sloane, Feb 23 2009

Or, read the following table by columns:
17,41,73,89,97,113,137,193,233,241,257,281,313,337,353,401,409,... (primes = = 1 mod 8)
3,11,19,43,59,67,83,107,131,139,163,179,211,227,251,283,307,331,... (primes == 3 mod 8)
5,13,29,37,53,61,101,109,149,157,173,181,197,229,269,277,293,317,... (primes == 5 mod 8)
7,23,31,47,71,79,103,127,151,167,191,199,223,239,263,271,311,359,... (primes == 7 mod 8)

			The first four primes congruent to (1,3,5,7) mod 8 are 17,3,5,7, hence a(1..4)=17,3,5,7;
The next four primes congruent to (1,3,5,7) mod 8 are 41,11,13,23, hence a(5..8)=41,11,13,23, etc.

Zak Seidov, Table of n, a(n) for n = 1..4000

Cf. A007519 - A007522, A042987.

s[i_]:=(c=0;a=2*i-1;Reap[Do[If[PrimeQ[a],c++;Sow[a]];If[c>99,Break[],a = a+8],{10^8}]][[2,1]]);Flatten[Transpose[Table[s[i],{i,4}]]]; (* Zak Seidov, Jan 16 2013 *)

A007528 Primes of the form 6k-1.

Original entry on oeis.org

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

A004771 a(n) = 8*n + 7. Or, numbers whose binary expansion ends in 111.

Original entry on oeis.org

7, 15, 23, 31, 39, 47, 55, 63, 71, 79, 87, 95, 103, 111, 119, 127, 135, 143, 151, 159, 167, 175, 183, 191, 199, 207, 215, 223, 231, 239, 247, 255, 263, 271, 279, 287, 295, 303, 311, 319, 327, 335, 343, 351, 359, 367, 375, 383, 391, 399, 407, 415, 423, 431

Offset: 0

N. J. A. Sloane

These numbers cannot be expressed as the sum of 3 squares. - Artur Jasinski, Nov 22 2006
These numbers cannot be perfect squares. - Cino Hilliard, Sep 03 2006
a(n-2), n >= 2, appears in the second column of triangle A239126 related to the Collatz problem. - Wolfdieter Lang, Mar 14 2014
The initial terms 7, 15, 23, 31 are the generating set for the rest of the sequence in the sense that, by Lagrange's Four Square Theorem, any number n of the form 8*k+7 can always be written as a sum of no fewer than four squares, and if n = a^2 + b^2 + c^2 + d^2, then (a mod 4)^2 + (b mod 4)^2 + (c mod 4)^2 + (d mod 4)^2 must be one of 7, 15, 23, 31. - Walter Kehowski, Jul 07 2014
Define a set of consecutive positive odd numbers {1, 3, 5, ..., 12*n + 9} and skip the number 6*n + 5. Then the contraharmonic mean of that set gives this sequence. For example, ContraharmonicMean[{1, 3, 7, 9}] = 7. - Hilko Koning, Aug 27 2018
Jacobi symbol (2, a(n)) = Kronecker symbol (a(n), 2) = 1. - Jianing Song, Aug 28 2018

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 246.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences
Leo Tavares, Illustration: Twin Square Frames
Leo Tavares, Illustration: Mid-line Hexagons
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 962
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A004767, A008590, A017077, A017101, A017137, A033954.
Cf. A007522 (primes), subsequence of A047522.
Cf. A056753, A227144, A227146, A239126.

List([0..60],n->8*n+7); # Muniru A Asiru, Aug 28 2018

a004771 = (+ 7) . (* 8)
a004771_list = [7, 15 ..]  -- Reinhard Zumkeller, Jan 29 2013

[8*n+7: n in [0..60]]; // Vincenzo Librandi, May 28 2011

A004771:=n->8*n+7; seq(A004771(n), n=0..100); # Wesley Ivan Hurt, Dec 22 2013

8 Range[0, 60] + 7 (* or *) Range[7, 500, 8] (* or *) Table[8 n + 7, {n, 0, 60}] (* Bruno Berselli, Dec 28 2016 *)

a(n)=8*n+7 \\ Charles R Greathouse IV, Sep 23 2012

O.g.f: (7 + x)/(1 - x)^2 = 8/(1 - x)^2 - 1/(1 - x). - R. J. Mathar, Nov 30 2007
a(n) = 2*a(n-1) - a(n-2) for n >= 2.  - Vincenzo Librandi, May 28 2011
A056753(a(n)) = 7. - Reinhard Zumkeller, Aug 23 2009
a(n) = t(t(t(n))), where t(i) = 2*i + 1.
a(n) = A004767(2*n+1), for n >= 0. See also A004767(2*n) = A017101(n). - Wolfdieter Lang, Feb 03 2022
From Elmo R. Oliveira, Apr 11 2024: (Start)
E.g.f.: exp(x)*(7 + 8*x).
a(n) = A033954(n+1) - A033954(n). (End)

A001132 Primes == +-1 (mod 8).

Original entry on oeis.org

7, 17, 23, 31, 41, 47, 71, 73, 79, 89, 97, 103, 113, 127, 137, 151, 167, 191, 193, 199, 223, 233, 239, 241, 257, 263, 271, 281, 311, 313, 337, 353, 359, 367, 383, 401, 409, 431, 433, 439, 449, 457, 463, 479, 487, 503, 521, 569, 577, 593, 599

Offset: 1

N. J. A. Sloane

Primes p such that 2 is a quadratic residue mod p.
Also primes p such that p divides 2^((p-1)/2) - 1. - Cino Hilliard, Sep 04 2004
A001132 is exactly formed by the prime numbers of A118905: in fact at first every prime p of A118905 is p = u^2 - v^2 + 2uv, with for example u odd and v even so that p - 1 = 4u'(u' + 1) + 4v'(2u' + 1 - v') when u = 2u' + 1 and v = 2v'. u'(u' + 1) is even and v'(2u' + 1 - v') is always even. At second hand if p = 8k +- 1, p has the shape x^2 - 2y^2; letting u = x - y and v = y, comes p = (x - y)^2 - y^2 + 2(x - y)y = u^2 - v^2 + 2uv so p is a sum of the two legs of a Pythagorean triangle. - Richard Choulet, Dec 16 2008
These are also the primes of form x^2 - 2y^2, excluding 2. See A038873. - Tito Piezas III, Dec 28 2008
Primes p such that p^2 mod 48 = 1. - Gary Detlefs, Dec 29 2011
Primes in A047522. - Reinhard Zumkeller, Jan 07 2012
This sequence gives the odd primes p which satisfy C(p, x = 0) = +1, where C(p, x) is the minimal polynomial of 2*cos(Pi/p) (see A187360). For the proof see a comment on C(n, 0) in A230075.  - Wolfdieter Lang, Oct 24 2013
Each a(n) corresponds to precisely one primitive Pythagorean triangle. For a proof see the W. Lang link, also for a table. See also the comment by Richard Choulet above, where the case u even and v odd has not been considered. - Wolfdieter Lang, Feb 17 2015
Primes p such that p^2 mod 16 = 1. - Vincenzo Librandi, May 23 2016
Rational primes that decompose in the field Q(sqrt(2)). - N. J. A. Sloane, Dec 26 2017

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
Ronald S. Irving, Integers, Polynomials, and Rings. New York: Springer-Verlag (2004): 274.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
C. Banderier, Calcul de (2/p)
Wolfdieter Lang, A001132: Primes +1 (mod 8) or -1 (mod 8) and Sum of Legs of Primitive Pythagorean Triangles.
Index entries for related sequences
Index to sequences related to decomposition of primes in quadratic fields

For primes p such that x^m = 2 (mod p) has a solution see A001132 (for m = 2), A040028 (m = 3), A040098 (m = 4), A040159 (m = 5), A040992 (m = 6), A042966 (m = 7), A045315 (m = 8), A049596 (m = 9), A049542 (m = 10) - A049595 (m = 63). Jeff Lagarias (lagarias(AT)umich.edu) points out that all these sequences are different, although this may not be apparent from looking just at the initial terms.
Agrees with A038873 except for initial term.
Cf. A118905, A010051.
Union of A007519 and A007522.

a001132 n = a001132_list !! (n-1)
a001132_list = [x | x <- a047522_list, a010051 x == 1]
-- Reinhard Zumkeller, Jan 07 2012

[p: p in PrimesUpTo (600) | p^2 mod 16 eq 1]; // Vincenzo Librandi, May 23 2016

seq(`if`(member(ithprime(n) mod 8, {1,7}),ithprime(n),NULL),n=1..109); # Nathaniel Johnston, Jun 26 2011
for n from 1 to 600 do if (ithprime(n)^2 mod 48 = 1) then print(ithprime(n)) fi od. # Gary Detlefs, Dec 29 2011

Select[Prime[Range[250]], MemberQ[{1, 7}, Mod[#, 8]] &]  (* Harvey P. Dale, Apr 29 2011 *)
Select[Union[8Range[100] - 1, 8Range[100] + 1], PrimeQ] (* Alonso del Arte, May 22 2016 *)

select(p->p%8==1 ||p%8==7, primes(100)) \\ Charles R Greathouse IV, May 18 2015

a(n) ~ 2n log n. - Charles R Greathouse IV, May 18 2015

A127576 Primes of the form 16n+15.

Original entry on oeis.org

31, 47, 79, 127, 191, 223, 239, 271, 367, 383, 431, 463, 479, 607, 719, 751, 863, 911, 991, 1039, 1087, 1103, 1151, 1231, 1279, 1327, 1423, 1439, 1471, 1487, 1567, 1583, 1663, 1759, 1823, 1871, 1951, 1999, 2063, 2111, 2143, 2207, 2239, 2287, 2351, 2383

Offset: 1

Artur Jasinski, Jan 19 2007

nonn

Subsequence of A007522. - R. J. Mathar, Jan 07 2009

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

Cf. A125169, A127575.

a = {}; Do[If[PrimeQ[16n + 15], AppendTo[a, 16n + 15]], {n, 1, 200}]; a
Select[16*Range[400]+15,PrimeQ] (* Harvey P. Dale, Feb 06 2013 *)

select(n->n%16==15, primes(100)) \\ Charles R Greathouse IV, Apr 28 2015

A035050 a(n) is the smallest k such that k*2^n + 1 is prime.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 3, 2, 1, 15, 12, 6, 3, 5, 4, 2, 1, 6, 3, 11, 7, 11, 25, 20, 10, 5, 7, 15, 12, 6, 3, 35, 18, 9, 12, 6, 3, 15, 10, 5, 6, 3, 9, 9, 15, 35, 19, 27, 15, 14, 7, 14, 7, 20, 10, 5, 27, 29, 54, 27, 31, 36, 18, 9, 12, 6, 3, 9, 31, 23, 39, 39, 40, 20, 10, 5, 58, 29, 15, 36, 18, 9, 13

Offset: 0

Labos Elemer

nonn

From Ulrich Krug (leuchtfeuer37(AT)gmx.de), Jun 05 2010: (Start)
If a(i) = 2 * m then a(i+1) = m.
Proof: (I) a(i) = 2*m, 2*m * 2^i + 1 = m*2^(i+1) + 1 prime, so a(i+1) <= m;
(II) if a(i+1) = m-d for an integer d > 0, (m-d) * 2^(i+1) + 1 = (2*m-2*d) * 2^i + 1 prime;
(2m-2d) < 2m contradiction to a(i) = 2 * m, d = 0.
(End)
Conjecture: for n > 0, a(n) = k < 2^n, so k*2^n + 1 is a Proth prime A080076. - Thomas Ordowski, Apr 13 2019

			a(3)=2 because 1*2^3 + 1 = 9 is composite, 2*2^3 + 1 = 17 is prime.
a(99)=219 because 2^99k + 1 is not prime for k=1,2,...,218. The first term which is not a composite number of this arithmetic progression is 2^99*219 + 1.

T. D. Noe, Table of n, a(n) for n = 0..1000
Gareth A. Jones and Alexander K. Zvonkin, Groups of prime degree and the Bateman-Horn conjecture, 2021.
Index entries for sequences related to primes in arithmetic progressions

Analogous case is A034693. Special subscripts (n's for a(n)=1) are the exponents of known Fermat primes: A000215. See also Fermat numbers A000051.

Cf. A007522, A057778, A080076, A085427, A087522, A126717, A127575, A127576, A127577, A127578, A127580, A127581, A127586.

sol:=[];m:=1; for n in [0..82] do k:=0; while not IsPrime(k*2^n+1) do k:=k+1; end while; sol[m]:=k; m:=m+1; end for; sol; // Marius A. Burtea, Jun 05 2019

a = {}; Do[k = 0; While[ ! PrimeQ[k 2^n + 1], k++ ]; AppendTo[a, k], {n, 0, 100}]; a (* Artur Jasinski *)
Table[Module[{k=1,n2=2^n},While[!PrimeQ[k*n2+1],k++];k],{n,0,90}] (* Harvey P. Dale, May 25 2024 *)

a(n) = {my(k = 1); while (! isprime(2^n*k+1), k++); k;}

a(n) << 19^n by Xylouris' improvement to Linnik's theorem. - Charles R Greathouse IV, Dec 10 2013

cp's OEIS Frontend

A254764 Fundamental positive solution x = x1(n) of the first class of the Pell equation x^2 - 2*y^2 = A007522(n), n >=1 (primes congruent to 7 mod 8).

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A254765 Fundamental positive solution y = y1(n) of the first class of the Pell equation x^2 - 2*y^2 = A007522(n), n >=1 (primes congruent to 7 mod 8).

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A186303 a(n) = ( A007522(n)+1 )/2.

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A186304 A007522(n)-2.

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A157115 Alternate terms of A007519, A007520, A007521, A007522.

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A007528 Primes of the form 6k-1.

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A004771 a(n) = 8*n + 7. Or, numbers whose binary expansion ends in 111.

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A001132 Primes == +-1 (mod 8).

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