A142925 -id:A142925 - cp's OEIS frontend

A002144 Pythagorean primes: primes of the form 4*k + 1.

5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461, 509, 521, 541, 557, 569, 577, 593, 601, 613, 617

Offset: 1

N. J. A. Sloane

Rational primes that decompose in the field Q(sqrt(-1)). - N. J. A. Sloane, Dec 25 2017
These are the prime terms of A009003.
-1 is a quadratic residue mod a prime p if and only if p is in this sequence.
Sin(a(n)*Pi/2) = 1 with Pi = 3.1415..., see A070750. - Reinhard Zumkeller, May 04 2002
If at least one of the odd primes p, q belongs to the sequence, then either both or neither of the congruences x^2 = p (mod q), x^2 = q (mod p) are solvable, according to Gauss reciprocity law. - Lekraj Beedassy, Jul 17 2003
Odd primes such that binomial(p-1, (p-1)/2) == 1 (mod p). - Benoit Cloitre, Feb 07 2004
Primes that are the hypotenuse of a right triangle with integer sides. The Pythagorean triple is {A002365(n), A002366(n), a(n)}.
Also, primes of the form a^k + b^k, k > 1. - Amarnath Murthy, Nov 17 2003
The square of a(n) is the average of two other squares. This fact gives rise to a class of monic polynomials x^2 + bx + c with b = a(n) that will factor over the integers regardless of the sign of c. See A114200. - Owen Mertens (owenmertens(AT)missouristate.edu), Nov 16 2005
Also such primes p that the last digit is always 1 for the Nexus numbers of form n^p - (n-1)^p. - Alexander Adamchuk, Aug 10 2006
The set of Pythagorean primes is a proper subset of the set of positive fundamental discriminants (A003658). - Paul Muljadi, Mar 28 2008
A079260(a(n)) = 1; complement of A137409. - Reinhard Zumkeller, Oct 11 2008
From Artur Jasinski, Dec 10 2008: (Start)
If we take 4 numbers: 1, A002314(n), A152676(n), A152680(n) then multiplication table modulo a(n) is isomorphic to the Latin square:
   1 2 3 4
   2 4 1 3
   3 1 4 2
   4 3 2 1
and isomorphic to the multiplication table of {1, i, -i, -1} where i is sqrt(-1), A152680(n) is isomorphic to -1, A002314(n) with i or -i and A152676(n) vice versa -i or i. 1, A002314(n), A152676(n), A152680(n) are subfield of Galois field [a(n)]. (End)
Primes p such that the arithmetic mean of divisors of p^3 is an integer. There are 2 sequences of such primes: this one and A002145. - Ctibor O. Zizka, Oct 20 2009
Equivalently, the primes p for which the smallest extension of F_p containing the square roots of unity (necessarily F_p) contains the 4th roots of unity. In this respect, the n = 2 case of a family of sequences: see n=3 (A129805) and n=5 (A172469). - Katherine E. Stange, Feb 03 2010
Subsequence of A007969. - Reinhard Zumkeller, Jun 18 2011
A151763(a(n)) = 1.
k^k - 1 is divisible by 4*k + 1 if 4*k + 1 is a prime (see Dickson reference). - Gary Detlefs, May 22 2013
Not only are the squares of these primes the sum of two nonzero squares, but the primes themselves are also. 2 is the only prime equal to the sum of two nonzero squares and whose square is not. 2 is therefore not a Pythagorean prime. - Jean-Christophe Hervé, Nov 10 2013
The statement that these primes are the sum of two nonzero squares follows from Fermat's theorem on the sum of two squares. - Jerzy R Borysowicz, Jan 02 2019
The decompositions of the prime and its square into two nonzero squares are unique. - Jean-Christophe Hervé, Nov 11 2013. See the Dickson reference, Vol. II, (B) on p. 227. - Wolfdieter Lang, Jan 13 2015
p^e for p prime of the form 4*k+1 and e >= 1 is the sum of 2 nonzero squares. - Jon Perry, Nov 23 2014
Primes p such that the area of the isosceles triangle of sides (p, p, q) for some integer q is an integer. - Michel Lagneau, Dec 31 2014
This is the set of all primes that are the average of two squares. - Richard R. Forberg, Mar 01 2015
Numbers k such that ((k-3)!!)^2 == -1 (mod k). - Thomas Ordowski, Jul 28 2016
This is a subsequence of primes of A004431 and also of A016813. - Bernard Schott, Apr 30 2022
In addition to the comment from Jean-Christophe Hervé, Nov 10 2013: All powers as well as the products of any of these primes are the sum of two nonzero squares. They are terms of A001481, which is closed under multiplication. - Klaus Purath, Nov 19 2023

			The following table shows the relationship between several closely related sequences:
Here p = A002144 = primes == 1 (mod 4), p = a^2+b^2 with a < b;
a = A002331, b = A002330, t_1 = ab/2 = A070151;
p^2 = c^2 + d^2 with c < d; c = A002366, d = A002365,
t_2 = 2ab = A145046, t_3 = b^2 - a^2 = A070079,
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
  ---------------------------------
   p  a  b  t_1  c   d t_2 t_3  t_4
  ---------------------------------
   5  1  2   1   3   4   4   3    6
  13  2  3   3   5  12  12   5   30
  17  1  4   2   8  15   8  15   60
  29  2  5   5  20  21  20  21  210
  37  1  6   3  12  35  12  35  210
  41  4  5  10   9  40  40   9  180
  53  2  7   7  28  45  28  45  630
  ...
a(7) = 53 = A002972(7)^2 + (2*A002973(7))^2 = 7^2 + (2*1)^2 = 49 + 4, and this is the only way. - _Wolfdieter Lang_, Jan 13 2015

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.
L. E. Dickson, "History of the Theory of Numbers", Chelsea Publishing Company, 1919, Vol I, page 386
L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 1920, p. 227.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 132.
M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; see p. 76.
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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 241, 243.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 90.

Zak Seidov, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972.
C. Banderier, Calcul de (-1/p).
J. Butcher, Mathematical Miniature 8: The Quadratic Residue Theorem, NZMS Newsletter, No. 75, April 1999.
Hing Lun Chan, Windmills of the minds: an algorithm for Fermat's Two Squares Theorem, arXiv:2112.02556 [cs.LO], 2021.
R. Chapman, Quadratic reciprocity.
A. David Christopher, A partition-theoretic proof of Fermat's Two Squares Theorem, Discrete Mathematics, Volume 339, Issue 4, 6 April 2016, Pages 1410-1411.
J. E. Ewell, A Simple Proof of Fermat's Two-Square Theorem, The American Mathematical Monthly, Vol. 90, No. 9 (Nov., 1983), pp. 635-637.
Bernard Frénicle de Bessy, Traité des triangles rectangles en nombres : dans lequel plusieurs belles propriétés de ces triangles sont démontrées par de nouveaux principes, Michalet, Paris (1676) pp. 0-116; see p. 44, Consequence II.
Bernard Frénicle de Bessy, Méthode pour trouver la solution des problèmes par les exclusions. Abrégé des combinaisons. Des Quarrez magiques, in "Divers ouvrages de mathématiques et de physique, par MM. de l'Académie royale des sciences", (1693) "Troisième exemple", pp. 17-26, see in particular p. 25.
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
D. & C. Hazzlewood, Quadratic Reciprocity.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Lucas Lacasa, Bartolome Luque, Ignacio Gómez, and Octavio Miramontes, On a Dynamical Approach to Some Prime Number Sequences, Entropy 20.2 (2018): 131, also arXiv:1802.08349 [math.NT], 2018.
R. C. Laubenbacher and D. J. Pengelley, Eisenstein's Misunderstood Geometric Proof of the Quadratic Reciprocity Theorem, In: Anderson M, Katz V, Wilson R, eds. Who Gave You the Epsilon?: And Other Tales of Mathematical History. Spectrum. Mathematical Association of America; 2009:309-312.
R. C. Laubenbacher and D. J. Pengelley, Gauss, Eisenstein and the 'third' proof of the Quadratic Reciprocity Theorem, The Mathematical Intelligencer 16, 67-72 (1994).
K. Matthews, Serret's algorithm based Server.
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. 521.
Carlos Rivera, Puzzle 968. Another property of primes 4m+1, The Prime Puzzles & Problems Connection.
D. Shanks, Review of "K. E. Kloss et al., Class number of primes of the form 4n+1", Math. Comp., 23 (1969), 213-214. [Annotated scanned preprint of review]
S. A. Shirali, A family portrait of primes-a case study in discrimination, Math. Mag. Vol. 70, No. 4 (Oct., 1997), pp. 263-272.
Rosemary Sullivan and Neil Watling, Independent divisibility pairs on the set of integers from 1 to n, INTEGERS 13 (2013) #A65.
Eric Weisstein's World of Mathematics, Wilson's Theorem.
Eric Weisstein's World of Mathematics, Pythagorean Triples.
Wikipedia, Quadratic reciprocity
Wolfram Research, The Gauss Reciprocity Law.
G. Xiao, Two squares.
D. Zagier, A One-Sentence Proof That Every Prime p == 1 (mod 4) Is a Sum of Two Squares, Am. Math. Monthly, Vol. 97, No. 2 (Feb 1990), p. 144. [From _Wolfdieter Lang_, Jan 17 2015 (thanks to Charles Nash)]
Index to sequences related to decomposition of primes in quadratic fields.

Cf. A002145, A002314, A002476, A002972, A002973, A003658, A004431, A007519, A010051, A016813, A076339, A094407.
Cf. A114200, A133870, A142925, A152676, A152680, A173330, A173331, A208177, A208178, A334912.
Cf. A004613 (multiplicative closure).
Apart from initial term, same as A002313.
For values of n see A005098.
Primes in A020668.

a002144 n = a002144_list !! (n-1)
a002144_list = filter ((== 1) . a010051) [1,5..]
-- Reinhard Zumkeller, Mar 06 2012, Feb 22 2011

[a: n in [0..200] | IsPrime(a) where a is 4*n + 1 ]; // Vincenzo Librandi, Nov 23 2014

a := []; for n from 1 to 500 do if isprime(4*n+1) then a := [op(a),4*n+1]; fi; od: A002144 := n->a[n];
# alternative
A002144 := proc(n)
    option remember ;
    local a;
    if n = 1 then
        5;
    else
        for a from procname(n-1)+4 by 4 do
            if isprime(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A002144(n),n=1..100) ; # R. J. Mathar, Jan 31 2024

Select[4*Range[140] + 1, PrimeQ[ # ] &] (* Stefan Steinerberger, Apr 16 2006 *)
Select[Prime[Range[150]],Mod[#,4]==1&] (* Harvey P. Dale, Jan 28 2021 *)

PARI
```
select(p->p%4==1,primes(1000))
```

A002144_next(p=A2144[#A2144])={until(isprime(p+=4),);p} /* NB: p must be of the form 4k+1. Beyond primelimit, this is *much* faster than forprime(p=...,, p%4==1 && return(p)). */
A2144=List(5); A002144(n)={while(#A2144A002144_next())); A2144[n]}
\\ M. F. Hasler, Jul 06 2024

from sympy import prime
A002144 = [n for n in (prime(x) for x in range(1,10**3)) if not (n-1) % 4]
# Chai Wah Wu, Sep 01 2014

from sympy import isprime
print(list(filter(isprime, range(1, 618, 4)))) # Michael S. Branicky, May 13 2021

def A002144_list(n): # returns all Pythagorean primes <= n
    return [x for x in prime_range(5,n+1) if x % 4 == 1]
A002144_list(617) # Peter Luschny, Sep 12 2012

Odd primes of form x^2 + y^2, (x=A002331, y=A002330, with x < y) or of form u^2 + 4*v^2, (u = A002972, v = A002973, with u odd). - Lekraj Beedassy, Jul 16 2004
p^2 - 1 = 12*Sum_{i = 0..floor(p/4)} floor(sqrt(i*p)) where p = a(n) = 4*n + 1. [Shirali]
a(n) = A000290(A002972(n)) + A000290(2*A002973(n)) = A000290(A002331(n+1)) + A000290(A002330(n+1)). - Reinhard Zumkeller, Feb 16 2010
a(n) = A002972(n)^2 + (2*A002973(n))^2, n >= 1. See the Jean-Christophe Hervé Nov 11 2013 comment. - Wolfdieter Lang, Jan 13 2015
a(n) = 4*A005098(n) + 1. - Zak Seidov, Sep 16 2018
From Vaclav Kotesovec, Apr 30 2020: (Start)
Product_{k>=1} (1 - 1/a(k)^2) = A088539.
Product_{k>=1} (1 + 1/a(k)^2) = A243380.
Product_{k>=1} (1 - 1/a(k)^3) = A334425.
Product_{k>=1} (1 + 1/a(k)^3) = A334424.
Product_{k>=1} (1 - 1/a(k)^4) = A334446.
Product_{k>=1} (1 + 1/a(k)^4) = A334445.
Product_{k>=1} (1 - 1/a(k)^5) = A334450.
Product_{k>=1} (1 + 1/a(k)^5) = A334449. (End)
From Vaclav Kotesovec, May 05 2020: (Start)
Product_{k>=1} (1 + 1/A002145(k)) / (1 + 1/a(k)) = Pi/(4*A064533^2) = 1.3447728438248695625516649942427635670667319092323632111110962...
Product_{k>=1} (1 - 1/A002145(k)) / (1 - 1/a(k)) = Pi/(8*A064533^2) = 0.6723864219124347812758324971213817835333659546161816055555481... (End)
Sum_{k >= 1} 1/a(k)^s = (1/2) * Sum_{n >= 1 odd numbers} moebius(n) * log((2*n*s)! * zeta(n*s) * abs(EulerE(n*s - 1)) / (Pi^(n*s) * 2^(2*n*s) * BernoulliB(2*n*s) * (2^(n*s) + 1) * (n*s - 1)!))/n, s >= 3 odd number. - Dimitris Valianatos, May 21 2020
Legendre symbol (-1, a(n)) = +1, for n >= 1. - Wolfdieter Lang, Mar 03 2021

A007519 Primes of form 8n+1, that is, primes congruent to 1 mod 8.

Original entry on oeis.org

17, 41, 73, 89, 97, 113, 137, 193, 233, 241, 257, 281, 313, 337, 353, 401, 409, 433, 449, 457, 521, 569, 577, 593, 601, 617, 641, 673, 761, 769, 809, 857, 881, 929, 937, 953, 977, 1009, 1033, 1049, 1097, 1129, 1153, 1193, 1201, 1217, 1249, 1289, 1297, 1321, 1361

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

Discriminant is 32, class is 2. Binary quadratic forms ax^2 + bxy + cy^2 have discriminant d = b^2 - 4ac and gcd(a, b, c) = 1.
Integers n (n > 9) of form 4k + 1 such that binomial(n-1, (n-1)/4) == 1 (mod n) - Benoit Cloitre, Feb 07 2004
Primes of the form x^2 + 8y^2. - T. D. Noe, May 07 2005
Also primes of the form x^2 + 16y^2. See A140633. - T. D. Noe, May 19 2008
Is this the same sequence as A141174?
Being a subset of A001132 and also a subset of A038873, this is also a subset of the primes of the form u^2 - 2v^2. - Tito Piezas III, Dec 28 2008
These primes p are only which possess the property: for every integer m from interval [0, p) with the Hamming distance D(m, p) = 2, there exists an integer h from (m, p) with D(m, h) = 2. - Vladimir Shevelev, Apr 18 2012
Primes p such that p XOR 6 = p + 6. - Brad Clardy, Jul 22 2012
Odd primes p such that -1 is a 4th power mod p. - Eric M. Schmidt, Mar 27 2014
There are infinitely many primes of this form. See Brubaker link. - Alonso del Arte, Jan 12 2017
These primes split in Z[sqrt(2)]. For example, 17 = (-1)(1 - 3*sqrt(2))(1 + 3*sqrt(2)). This is also true of primes of the form 8n - 1. - Alonso del Arte, Jan 26 2017

			a(1) = 17 = 2 * 8 + 1 = (10001)_2. All numbers m from [0, 17) with the Hamming distance D(m, 17) = 2 are 0, 3, 5, 9. For m = 0, we can take h = 3, since 3 is drawn from (0, 17) and D(0, 3) = 2; for m = 3, we can take h = 5, since 5 from (3, 17) and D(3, 5) = 2; for m = 5, we can take h = 6, since 6 from (5, 17) and D(5, 6) = 2; for m = 9, we can take h = 10, since 10 is drawn from (9, 17) and D(9, 10) = 2. - _Vladimir Shevelev_, Apr 18 2012

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.
Z. I. Borevich and I. R. Shafarevich, Number Theory.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 261.

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].
Ben Brubaker, 18.781, Fall 2007 Problem Set 5: Solutions to Selected Problems, MIT (2007).
Peter Luschny, Binary Quadratic Forms
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. 521.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Subsequence of A017077 and of A038873.
Cf. A139643. Complement in primes of A154264. Cf. A042987.
Cf. A065091, A002144, A094407, A133870, A142925, A208177, A208178, A076339.
Cf. A038872 (d = 5). A038873 (d = 8). A068228, A141123 (d = 12). A038883 (d = 13). A038889 (d = 17). A141111, A141112 (d = 65).
Cf. also A242663.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

a007519 n = a007519_list !! (n-1)
a007519_list = filter ((== 1) . a010051) [1,9..]
-- Reinhard Zumkeller, Mar 06 2012

[p: p in PrimesUpTo(2000) | p mod 8 eq 1 ]; // Vincenzo Librandi, Aug 21 2012

Select[1 + 8 Range@ 170, PrimeQ] (* Robert G. Wilson v *)

forprime(p=2,1e4,if(p%8==1,print1(p", "))) \\ Charles R Greathouse IV, Jun 16 2011

forprimestep(p=17,10^4,8, print1(p", ")) \\ Charles R Greathouse IV, Jul 17 2024

lista(nn)= my(vpr = []); for (x = 0, nn, y = 0; while ((v = x^2+6*x*y+y^2) < nn, if (isprime(v), if (! vecsearch(vpr, v), vpr = concat(vpr, v); vpr = vecsort(vpr););); y++;);); vpr; \\ Michel Marcus, Feb 01 2014

A007519_upto(N, start=1)=select(t->t%8==1,primes([start,N]))
#A7519=A007519_upto(10^5)
A007519(n)={while(#A7519A007519_upto(N*3\2, N+1))); A7519[n]} \\ M. F. Hasler, May 22 2025

# uses[binaryQF]
# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([1, 4, -4])
print(Q.represented_positives(1361, 'prime'))  # Peter Luschny, Jan 26 2017

A094407 Primes of the form 16n+1.

Original entry on oeis.org

17, 97, 113, 193, 241, 257, 337, 353, 401, 433, 449, 577, 593, 641, 673, 769, 881, 929, 977, 1009, 1153, 1201, 1217, 1249, 1297, 1361, 1409, 1489, 1553, 1601, 1697, 1777, 1873, 1889, 2017, 2081, 2113, 2129, 2161, 2273, 2417, 2593, 2609, 2657, 2689, 2753

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Jun 03 2004

Subsequence of A007519 (primes of form 8n+1). - Zak Seidov, May 16 2012
Primes p such that p XOR 14 = p + 14. - Brad Clardy, Jul 23 2012
A prime of the form 16n+1 is represented either by both x^2+32y^2 and x^2+64y^2 or by neither (see Kaplansky link). - Michel Marcus, Dec 23 2012
Odd primes p such that -1 is an 8th power mod p. - Eric M. Schmidt, Mar 27 2014

T. D. Noe, Table of n, a(n) for n=1..1000
C, Caldwell, Prime test.
Irving Kaplansky, The forms x+32y^2 and x+64y^2, Proc. Amer. Math. Soc. 131 (2003), 2299-2300

Primes congruent to k mod 16: A094407, A091968, A127589, A141194, A105126, A141195, A141196, A127576.

Cf. A065091, A002144, A007519, A133870, A142925, A208177, A208178, A076339.

a094407 n = a094407_list !! (n-1)
a094407_list = filter ((== 1) . a010051) [1,17..]
-- Reinhard Zumkeller, Mar 06 2012

p:=proc(n) if isprime(16*n+1)=true then 16*n+1 else fi end:seq(p(n),n=1..200); # Emeric Deutsch, Dec 23 2004

lst={};Do[p=16*n+1;If[PrimeQ[p],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 26 2009 *)
Select[16*Range[200]+1,PrimeQ] (* Harvey P. Dale, Nov 04 2017 *)

More terms from Emeric Deutsch, Dec 23 2004

A133870 Primes of the form 32*n + 1.

Original entry on oeis.org

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

Zak Seidov, Sep 27 2007

Corresponding n's: 3, 6, 8, 11, 14, 18, 20, 21, 24, 29, 36, 38, 39, ... (A133869).
These primes p are the only ones with the property that for every integer m from interval [0,p) with the Hamming  distance D(m,p) = 2 or 3, there exists an integer h from (m,p) with D(m,h) = D(m,p). - Vladimir Shevelev, Apr 19 2012
Primes p such that p XOR 30 = p + 30. - Brad Clardy, Jul 22 2012
Odd primes p such that -1 is a 16th power mod p. - Eric M. Schmidt, Mar 27 2014

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

Cf. A000040, A133869; A065091, A002144, A007519, A094407, A142925, A208177, A208178, A076339.

a133870 n = a133870_list !! (n-1)
a133870_list = filter ((== 1) . a010051) [1,33..]
-- Reinhard Zumkeller, Mar 06 2012

[p: p in PrimesUpTo(12000) | p mod 32 eq 1 ]; // Vincenzo Librandi, Aug 18 2012

Select[32*Range[175] + 1, PrimeQ] (* Alonso del Arte, Jul 24 2012 *)
Select[Prime[Range[4000]],MemberQ[{1},Mod[#,32]]&] (* Vincenzo Librandi, Aug 18 2012 *)

A076339 Primes of the form 512*k+1.

Original entry on oeis.org

7681, 10753, 11777, 12289, 13313, 15361, 17921, 18433, 19457, 23041, 25601, 26113, 32257, 36353, 37889, 39937, 40961, 45569, 50177, 51713, 58369, 59393, 61441, 64513, 65537, 67073, 70657, 76289, 76801, 79873, 80897, 81409, 83969

Offset: 1

Reinhard Zumkeller, Oct 07 2002

Odd primes p such that -1 is a 256th power mod p. - Eric M. Schmidt, Mar 27 2014

			A076338(15) = 512*15+1 = a(1) = 7681 = A000040(974);
A076338(21) = 512*21+1 = a(2) = 10753 = A000040(1311);
a(38) - a(37) = 95233 - 87553 = 7680 = a(1)-1.

M. Kraitchik, Theorie des Nombres, Gauthier-Villars (I. 1922, II. 1929).
M. Kraitchik, Recherches sur la theorie des nombres, Gauthier-Villars (1924).

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

Cf. A076338, A065091, A002144, A007519, A094407, A133870, A142925, A208177, A208178.

a076339 n = a076339_list !! (n-1)
a076339_list = filter ((== 1) . a010051) [1,513..]
-- Reinhard Zumkeller, Mar 06 2012

Select[512 Range[164] + 1, PrimeQ] (* Bruno Berselli, Feb 23 2012 *)

forprimestep(p=7681,83969,512, print1(p", ")) \\ Charles R Greathouse IV, Nov 01 2022

a(n) ~ 256n log n. - Charles R Greathouse IV, Nov 01 2022

A208177 Primes of the form 128*k + 1.

Original entry on oeis.org

257, 641, 769, 1153, 1409, 2689, 3329, 3457, 4481, 4993, 6529, 7297, 7681, 7937, 9473, 9601, 9857, 10369, 10753, 11393, 11777, 12161, 12289, 13313, 13441, 13697, 14081, 14593, 15233, 15361, 16001, 17921, 18049, 18433, 19073, 19457, 19841, 20353, 21121, 21377

Offset: 1

Bruno Berselli, Feb 25 2012

Odd primes p such that -1 is a 64th 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, A208178, A076339.

a208177 n = a208177_list !! (n-1)
a208177_list = filter ((== 1) . a010051) [1,129..]
-- Reinhard Zumkeller, Mar 06 2012

[ p: p in PrimesUpTo(22000) | p mod 128 eq 1 ];

Mathematica
```
Select[128 Range[167] + 1, PrimeQ]
```

for(n=1,167,r=128*n+1; if(isprime(r), print1(r", ")));

a(n) ~ 64n log n. - Charles R Greathouse IV, Nov 01 2022

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

A376349 Number of isomorphism classes k of groups G of order p*2^n when G contains a unique Sylow p subgroup and the maximal 2^m dividing p-1 is such that 2^m >= 2^n.

Original entry on oeis.org

1, 2, 5, 15, 54, 247, 1684, 21820, 1118964

Offset: 0

Miles Englezou, Sep 19 2024

A Sylow p subgroup is a subgroup of order p^r that necessarily exists when r is a maximal power of p. It is not necessarily unique, but when it is unique it is normal in G.
The condition that G of order p*2^n contains a unique Sylow p subgroup places an upper bound on the number of isomorphism classes of G; it is equivalent to stating that the minimal 2^r such that 2^r == 1 (mod p) be such that 2^r > 2^n. The condition that the maximal 2^m dividing p-1, i.e. for p == 1 (mod 2^m), is such that 2^m >= 2^n ensures a lower bound which is equal to the upper bound. See the Miles Englezou link for a proof.
If we relax the two conditions and just consider an arbitrary odd prime p and the number of isomorphism classes for |G| = p*2^n, it is likely that the set of such numbers is unique to p. Since every odd prime has a minimal 2^r such that 2^r == 1 (mod p) (a consequence of Fermat's little theorem), when 2^r = 2^n for |G| = p*2^n, the number of isomorphism classes will differ from a(n) due to the existence of groups where the Sylow p subgroup is not unique.

			a(2) = 5 since D_(p*2^2), C_(p*2^2), C_(p*2^1) x C_2, and two semidirect products C_p : C_4 are all the groups of order p*2^2 for p satisfying the two conditions.
Table showing minimal 2^r and maximal 2^m (as defined in the Comments) for some primes:
---------------------------------------------------------------------------
p |      Minimal 2^r == 1 (mod p)       |   Maximal 2^m, p == 1 (mod 2^m)  |
---------------------------------------------------------------------------
2 |             2^0  = 1                |              2^0 = 1             |
3 |             2^2  = 4                |              2^1 = 2             |
5 |             2^4  = 16               |              2^2 = 4             |
7 |             2^3  = 8                |              2^1 = 2             |
11|             2^10 = 1024             |              2^1 = 2             |
13|             2^12 = 4096             |              2^2 = 4             |
17|             2^8  = 256              |              2^4 = 16            |
19|             2^18 = 262144           |              2^1 = 2             |
23|             2^11 = 2048             |              2^1 = 2             |
29|             2^28 = 268435456        |              2^2 = 4             |
31|             2^5  = 32               |              2^1 = 2             |
37|             2^36 = 68719476736      |              2^2 = 4             |
---------------------------------------------------------------------------
Table of primes satisfying 2^r > 2^n, and 2^m >= 2^n:
-------------------------------------------------------------------------------
   2^n   |                          primes                           |   a(n)  |
-------------------------------------------------------------------------------
2^0 = 1  |  all primes                                    = A000040  | 1       |
2^1 = 2  |  all primes > 2                                = A065091  | 2       |
2^2 = 4  |  5, 13, 17, 29, 37, 41, 53, ...                = A002144  | 5       |
2^3 = 8  |  17, 41, 73, 89, 97, 113, 137, ...             = A007519  | 15      |
2^4 = 16 |  17, 97, 113, 193, 241, 257, 337 ...           = A094407  | 54      |
2^5 = 32 |  97, 193, 257, 353, 449, 577, 641, ...         = A133870  | 247     |
2^6 = 64 |  193, 257, 449, 577, 641, 769, 1153, ...       = A142925  | 1684    |
2^7 = 128|  257, 641, 769, 1153, 1409, 2689, 3329, ...    = A208177  | 21820   |
2^8 = 256|  257, 769, 3329, 7937, 9473, 14081, 14593 ...  = A105131  | 1118964 |
-------------------------------------------------------------------------------

Miles Englezou, Proof

Cf. A000001, A000040, A065091, A002144, A007519, A094407, A133870, A142925, A208177, A105131, A139669, A014664, A023506, A058384.

S:=[];
for i in [0..8] do
    n:=7681*2^i; # 7681 is an appropriate prime for reproducing up to a(8)
    S:=Concatenation(S,[NrSmallGroups(n)]);
od;
Print(S);

a(n) = A000001(p*2^(n)) for every p satisfying the two conditions mentioned in Comments.

cp's OEIS Frontend

A218163 a(n) is the smallest positive integer k such that k^32 + 1 == 0 mod p, where p is the n-th prime of the form 1 + 64*b (see A142925).

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A002144 Pythagorean primes: primes of the form 4*k + 1.

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A007519 Primes of form 8n+1, that is, primes congruent to 1 mod 8.

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A094407 Primes of the form 16n+1.

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A133870 Primes of the form 32*n + 1.

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A076339 Primes of the form 512*k+1.

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