A002144 -id:A002144 - cp's OEIS frontend

A014442 Largest prime factor of n^2 + 1.

2, 5, 5, 17, 13, 37, 5, 13, 41, 101, 61, 29, 17, 197, 113, 257, 29, 13, 181, 401, 17, 97, 53, 577, 313, 677, 73, 157, 421, 53, 37, 41, 109, 89, 613, 1297, 137, 17, 761, 1601, 29, 353, 37, 149, 1013, 73, 17, 461, 1201, 61, 1301, 541, 281, 2917, 89, 3137, 13, 673

Offset: 1

Glen Burch (gburch(AT)erols.com)

All a(n) except for a(1) = 2 are the Pythagorean primes, i.e., primes of form 4k+1. Conjecture: every Pythagorean prime appears in a(n) at least once.
Problem 11831 [Ozols 2015] is to prove that lim inf a(n)/n is zero. - Michael Somos, May 11 2015
From Michael Kaltman, Jun 10 2015: (Start)
For all numbers k in A256011, a(k) < k.
Conjecture: every Pythagorean prime p appears exactly two times among the first p integers of the sequence. Further: if a(i) = a(j) = p and both i and j are less than p (and i is not equal to j), then i + j = p and ij == 1 (mod p). [If a(k) = p as well, then k > p; in fact, k is in A256011.] Two examples: a(2) = a(3) = 5, with 2+3 = 5 and 2*3 = 6 == 1 (mod 5); a(4) = a(13) = 17, with 4+13 = 17 and 4*13 = 52 == 1 (mod 17).
(End)
The conjecture is true. If p is a Pythagorean prime, -1 is a quadratic residue mod p. Then -1 has exactly two square roots mod p, i.e., there are exactly two integers x,y with 1 <= x,y <= p-1 such that x^2 == y^2 == -1 (mod p), i.e., p divides x^2+1 and y^2+1, and moreover y == -x (mod p) so x + y = p, and x*y == -x^2 == 1 (mod p). Any other prime factor q of x^2 + 1 must divide (x^2+1)/p, and since x^2+1 < p^2 we have q < p, so a(x) = p and similarly a(y) = p. - Robert Israel, Jun 11 2015
Conjecture: if n is even and a(n) > n, then n+a(n) is in A256011. Examples: 2+a(2) = 2+5 = 7, 4+a(4) = 4+17 = 21, 6+a(6) = 6+37 = 43, and so on. Note that 18+a(18) is NOT in A256011, but 18 itself is. - Michael Kaltman, Jun 13 2015
This is also true. Suppose A = a(n) > n. n^2+1 is odd so A is an odd prime; n^2 + 1 = A *B with B < A also odd. Then (A+n)^2 + 1 = A*(A+2*n+B) and A+2*n+B is even. The largest prime factor of A+2*n+B is thus at most (A+2*n+B)/2 < A + n, while A < A + n as well. - Robert Israel, Jun 17 2015
Størmer shows that a(n) tends to infinity with n. Chowla shows that a(n) >> log log n.  Schinzel shows that lim inf a(n)/log log n >= 4 and, using lower bounds for linear forms of logarithms, this inequality can be generalized for general quadratic polynomials, with 2 replaced by 4/7 for irreducible ones and 2/7 for reducible ones. - Tomohiro Yamada, Apr 15 2017
According to Hooley, an unpublished manuscript of Chebyshev contains the result that a(n)/n is unbounded which was first published and fully proved by Markov. - Charles R Greathouse IV, Oct 27 2018
Note that a(n) is the largest prime p such that n^(p+1) == -1 (mod p). - Thomas Ordowski, Nov 08 2019

A. A. Markov, Über die Primteiler der Zahlen von der Form 1+4x^2, Bulletin de l'Académie impériale des sciences de St.-Pétersbourg 3 (1895), pp. 55-59.
H. Rademacher, Lectures on Elementary Number Theory, pp. 33-38.

T. D. Noe, Table of n, a(n) for n = 1..10000
S. Chowla, The greatest prime factor of x^2 + 1, J. London Math. Soc. 10 (1935), 117--120.
J. M. Deshouillers and H. Iwaniec, On the greatest prime factor of n^2+1, Annales de l'institut Fourier, 32 no. 4 (1982), p. 1-11.
Christopher Hooley, On the greatest prime factor of a quadratic polynomial, Acta Mathematica 117 (1967), pp. 281-299.
Jori Merikoski, Largest prime factor of n^2+1, arXiv:1908.08816 [math.NT], 2019.
R. Ozols, Problem 11831, The American Mathematical Monthly, Vol. 122, No. 4 (April 2015), page 390
A. Schinzel, On two theorems of Gelfond and some of their applications, Acta Arith. 13 (1967), 177--236.
Carl Størmer, Quelques théorèmes sur l'équation de Pell x^2 - Dy^2 = +-1 et leurs applications (in French), Skrifter Videnskabs-selskabet (Christiania), Mat.-Naturv. Kl. I (2), 48 pp.

Includes primes from A002496.
Cf. A002144 (Pythagorean primes: primes of form 4n+1).
Cf. A256011.
Cf. A076605 (largest prime factor of n^2 - 1).

List([1..60],n->Reversed(Factors(n^2+1))[1]); # Muniru A Asiru, Oct 27 2018

[Maximum(PrimeDivisors(n^2+1)): n in [1..60]]; // Vincenzo Librandi, Jun 17 2015

seq(max(numtheory:-factorset(n^2+1)),n=1..100) ; # Robert Israel, Jun 11 2015

Table[FactorInteger[n^2+1,FactorComplete->True][[ -1,1]],{n,5!}] ..and/or.. Table[Last[Table[ #[[1]]]&/@FactorInteger[n^2+1]],{n,5!}] ..and/or.. PrimeFactors[n_]:=Flatten[Table[ #[[1]],{1}]&/@FactorInteger[n]]; Table[PrimeFactors[n^2+1][[ -1]],{n,5!}] (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)
a[ n_] := If[ n < 1, 0, FactorInteger[n n + 1][[All, 1]] // Last]; (* Michael Somos, May 11 2015 *)
Table[FactorInteger[n^2 + 1][[-1, 1]], {n, 80}] (* Vincenzo Librandi, Jun 17 2015 *)

largeasqp1(m) = { for(a=1,m, y=a^2 + 1; f = factor(y); v = component(f,1); v1 = v[length(v)]; print1(v1",") ) } \\ Cino Hilliard, Jun 12 2004

{a(n) = if( n<1, 0, Vecrev(factor(n*n + 1)[,1])[1])}; /* Michael Somos, May 11 2015 */

a(n) = A006530(1+n^2). - R. J. Mathar, Jan 28 2017

A027862 Primes of the form j^2 + (j+1)^2.

Original entry on oeis.org

5, 13, 41, 61, 113, 181, 313, 421, 613, 761, 1013, 1201, 1301, 1741, 1861, 2113, 2381, 2521, 3121, 3613, 4513, 5101, 7321, 8581, 9661, 9941, 10513, 12641, 13613, 14281, 14621, 15313, 16381, 19013, 19801, 20201, 21013, 21841, 23981, 24421, 26681

Offset: 1

Patrick De Geest

Also, primes of the form 4*k+1 which are the hypotenuse of one and only one right triangle with integral legs. - Cino Hilliard, Mar 16 2003
Centered square primes (i.e., prime terms of centered squares A001844). - Lekraj Beedassy, Jan 21 2005
Primes of the form 2*k*(k-1)+1. - Juri-Stepan Gerasimov, Apr 27 2010
Equivalently, primes of the form (m^2+1)/2 (take m=2*j+1). These primes a(n) have nontrivial solutions of x^2 == 1 (Modd a(n)) given by x=x(n)=A002731(n). For Modd n see a comment on A203571. See also A206549 for such solutions for primes of the form 4*k+1, given in A002144.
  E.g., a(3)=41, A002731(3)=9, 9^2=81, floor(81/41)=1 (odd),
  -81 = -2*41 + 1 == 1 (mod 2*41), hence 9^2 == 1 (Modd 41). - Wolfdieter Lang, Feb 24 2012
Also primes of the form 4*k+1 that are the smallest side length of one and only one integer Soddyian triangle (see A230812). - Frank M Jackson, Mar 13 2014
Also, primes of the form (m^2+1)/2. - Zak Seidov, May 01 2014
Note that ((2n+1)^2 + 1)/2 = n^2 + (n+1)^2. - Thomas Ordowski, May 25 2015
Primes p such that 2p-1 is a square. - Thomas Ordowski, Aug 27 2016
Primes in the main diagonal of A000027 when represented as an array read by antidiagonals. - Clark Kimberling, Mar 12 2023
The diophantine equation x^2 + ... + (x + r)^2 = p may be rewritten to A*x^2 + B*x + C = p, where A = (r + 1), B = r*(r + 1), C = r*(r + 1)*(2*r + 1)/6. If gcd(A, B, C) > 1 no solution for a prime p exists. The gcd(A, B, C) = 1 holds only for r = 1, 2, 5 (gcd is the greatest common divisor). For r = 1 we have x^2 + (x + 1)^2 = p, thus for x from A027861 we calculate primes p from A027862. For r = 2 we have x^2 + (x + 1)^2 + (x + 2)^2 = p, thus for x from A027863 we calculate primes p from A027864. For r = 5 we have x^2 + ... + (x + 5)^2 = p, thus for x from A027866 we calculate primes p from A027867. - Ctibor O. Zizka, Oct 04 2023

			13 is in the sequence because it is prime and 13 = 2^2 + 3^2. - _Michael B. Porter_, Aug 27 2016

D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc. Boston, MA, 1976, p. 271.
Morris Kline, Mathematical Thought from Ancient to Modern Times, 1972. pp. 275.

T. D. Noe and Zak Seidov, Table of n, a(n) for n = 1..10000
Patrick De Geest, World!Of Numbers
Daniel Shanks, An analytic criterion for the existence of infinitely many primes of the form 1/2 * (n^2 + 1), Illinois Journal of Mathematics 8:3 (1964), p. 377-379.
Wacław Sierpiński, Sur les nombres triangulaires qui sont sommes de deux nombres triangulaires, Elem. Math., 17 (1962), pp. 63-65.
Panayiotis G. Tsangaris, A sieve for all primes of the form x^2 + (x+1)^2, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae, 25 (1998), pp. 39-53.

Primes p such that A079887(p) = 1.
Cf. A002731 (m values), A027861 (j values), A091277 (prime indices).
Subsequence of A002144 (p=4k+1).
Cf. A001844 (centered squares), A027863, A027864, A027866, A027867, A203571, A206549, A230812.

[ a: n in [0..150] | IsPrime(a) where a is n^2+(n+1)^2 ]; // Vincenzo Librandi, Dec 18 2010

Select[Table[n^2+(n+1)^2,{n,200}],PrimeQ] (* Harvey P. Dale, Aug 22 2012 *)
Select[Total/@Partition[Range[200]^2,2,1],PrimeQ] (* Harvey P. Dale, Apr 20 2016 *)

je=[]; for(n=1,500, if(isprime(n^2+(n+1)^2),je=concat(je,n^2+(n+1)^2))); je

fermat(n) = { for(x=1,n, y=2*x*(x+1)+1; if(isprime(y),print1(y" ")) ) }

a(n) = ((A002731(n)^2 - 1)/2) + 1. - Torlach Rush, Mar 14 2014

a(n) = (A002731(n)^2 + 1)/2. - Zak Seidov, May 01 2014

More terms from Cino Hilliard, Mar 16 2003

A004613 Numbers that are divisible only by primes congruent to 1 mod 4.

Original entry on oeis.org

1, 5, 13, 17, 25, 29, 37, 41, 53, 61, 65, 73, 85, 89, 97, 101, 109, 113, 125, 137, 145, 149, 157, 169, 173, 181, 185, 193, 197, 205, 221, 229, 233, 241, 257, 265, 269, 277, 281, 289, 293, 305, 313, 317, 325, 337, 349, 353, 365, 373, 377, 389, 397, 401, 409, 421

Offset: 1

N. J. A. Sloane

Also gives solutions z to x^2+y^2=z^4 with gcd(x,y,z)=1 and x,y,z positive. - John Sillcox (johnsillcox(AT)hotmail.com), Feb 20 2004
A065338(a(n)) = 1. - Reinhard Zumkeller, Jul 10 2010
Product_{k=1..A001221(a(n))} A079260(A027748(a(n),k)) = 1. - Reinhard Zumkeller, Jan 07 2013
A062327(a(n)) = A000005(a(n))^2. (These are the only numbers that satisfy this equation.) - Benedikt Otten, May 22 2013
Numbers that are positive integer divisors of 1 + 4*x^2 where x is a positive integer. - Michael Somos, Jul 26 2013
Numbers k such that there is a "knight's move" of Euclidean distance sqrt(k) which allows the whole of the 2D lattice to be reached. For example, a knight which travels 4 units in any direction and then 1 unit at right angles to the first direction moves a distance sqrt(17) for each move. This knight can reach every square of an infinite chessboard.
Also 1/7 of the area of the n-th largest octagon with angles 3*Pi/4, along the perimeter of which there are only 8 nodes of the square lattice - at its vertices. - Alexander M. Domashenko, Feb 21 2024
Sequence closed under multiplication. Odd values of A031396 and their powers. These are the only numbers m that satisfy the Pell equation (k*x)^2 - D*(m*y)^2 = -1. - Klaus Purath, May 12 2025

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

T. D. Noe, Table of n, a(n) for n = 1..10000
David Broadhurst and Wadim Zudilin, A magnetic double integral, arXiv:1708.02381 [math.NT], 2017. See p. 7
Code Golf Stack Exchange, pseudonymous Stack Exchange user 'trichoplax', N-movers: How much of the infinite board can I reach?

Subsequence of A000404; A002144 is a subsequence. Essentially same as A008846.

Cf. A004614.

a004613 n = a004613_list !! (n-1)
a004613_list = filter (all (== 1) . map a079260 . a027748_row) [1..]
-- Reinhard Zumkeller, Jan 07 2013

[n: n in [1..500] | forall{d: d in PrimeDivisors(n) | d mod 4 eq 1}]; // Vincenzo Librandi, Aug 21 2012

isA004613 := proc(n)
    local p;
    for p in numtheory[factorset](n) do
        if modp(p,4) <> 1 then
            return false;
        end if;
    end do:
    true;
end proc:
for n from 1 to 200 do
    if isA004613(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Nov 17 2014
# second Maple program:
q:= n-> andmap(i-> irem(i[1], 4)=1, ifactors(n)[2]):
select(q, [$1..500])[];  # Alois P. Heinz, Jan 13 2024

ok[1] = True; ok[n_] := And @@ (Mod[#, 4] == 1 &) /@ FactorInteger[n][[All, 1]]; Select[Range[421], ok] (* Jean-François Alcover, May 05 2011 *)
Select[Range[500],Union[Mod[#,4]&/@(FactorInteger[#][[All,1]])]=={1}&] (* Harvey P. Dale, Mar 08 2017 *)

for(n=1,1000,if(sumdiv(n,d,isprime(d)*if((d-1)%4,1,0))==0,print1(n,",")))

is(n)=n%4==1 && factorback(factor(n)[,1]%4)==1 \\ Charles R Greathouse IV, Sep 19 2016

Numbers of the form x^2 + y^2 where x is even, y is odd and gcd(x, y) = 1.

A051222 Numbers k such that Bernoulli number B_{k} has denominator 6.

Original entry on oeis.org

2, 14, 26, 34, 38, 62, 74, 86, 94, 98, 118, 122, 134, 142, 146, 158, 182, 194, 202, 206, 214, 218, 254, 266, 274, 278, 298, 302, 314, 326, 334, 338, 362, 386, 394, 398, 422, 434, 446, 454, 458, 482, 494, 514, 518, 526, 538, 542, 554, 566, 578

Offset: 1

N. J. A. Sloane

Alternative definition: let D(m) = set of divisors of m; sequence gives n such that the set 1 + D(n) contains only two primes, 2 and 3. E.g., n=98: D(98)={1,2,7,15,49,98}, 1+D = {2,3,8,16,50,99} of which only 2 terms are prime numbers: {2,3}. Observation by Labos Elemer, Jun 24 2002. This is a consequence of the von Staudt-Clausen theorem. - N. J. A. Sloane, Jan 04 2004
The fraction of Bernoulli numbers with denominator 6 is roughly 1/6, see Erdős-Wagstaff. But calculations by H. Cohen and G. Tenenbaum suggest that the fraction is closer to 1/7 (posting to Number Theory List around Dec 20 2005).
Simon Plouffe reports (Feb 13 2007) that at B_{9083002} the proportion is 0.151848915149418661363281... and still decreasing very slowly.
In his PhD thesis at the University of Illinois (see reference), Richard Sunseri proved that a higher proportion of Bernoulli denominators equal 6 than any other value.
Rado showed that for a given Bernoulli number B_n there exist infinitely many Bernoulli numbers B_m having the same denominator. As a special case, if n = 2p where p is an odd prime p == 1 (mod 3), then the denominator of the Bernoulli number B_n equals 6. - Bernd C. Kellner, Mar 21 2018
Conjecture: When the expression (p+q^b)/2 is required to be prime, p is prime, and q is a prime >=5, then all p values are prime congruent to 1 (mod 12) (A068228), if and only if the exponent b is a member of this set. - Richard R. Forberg, Apr 07 2025
There are additional exponential expressions conjectured for generating each of several known prime subsequences (e.g., Pythagorean primes, A002144) where the sequence is invariant to the exponent, if and only if the exponent is a member of this set.  See Forberg link. - Richard R. Forberg, Apr 25 2025

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 75.
C. J. Moreno and S. S. Wagstaff, Sums of Squares of Integers, CRC Press, 2005, Sect. 3.9.
H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1, p. 10.

T. D. Noe, Table of n, a(n) for n = 1..1000
Paul Erdős and Samuel S. Wagstaff, Jr., The fractional parts of the Bernoulli numbers, Illinois J. Math. 24 (1980), pp. 104-112, MR 81c:10064.
Richard R. Forberg, A051222 and Exponential Expressions the Reproduce Certain Prime Subsets, 2025.
K. L. Jensen, Om talteoretiske Egenskaber ved de Bernoulliske Tal, Nyt Tidskrift für Math. Afdeling B 28 (1915), pp. 73-83.
R. Rado, A note on the Bernoullian numbers, J. London Math. Soc. 9 (1934) 88-90.
Richard Sunseri, Zeros of p-adic L-functions and densities relating to Bernoulli numbers, PhD thesis, University of Illinois, 1979.
Index entries for sequences related to Bernoulli numbers.

Except for 2, all terms are even nontotient numbers. Proper subset of A005277: e.g., 50 and 90 are not here. - Labos Elemer
A112772 is a subsequence. - Bernd C. Kellner, Mar 21 2018
Cf. A045979, A000005, A067513, A002202, A005277.

di[x_] := Divisors[x]
dp[x_] := Part[di[x], Flatten[Position[PrimeQ[1+di[x]], True]]]+1
Do[s=Length[dp[n]]; If[Equal[s, 2], Print[n]], {n, 1, 10000}] (* Labos Elemer *)
Do[s=Denominator[BernoulliB[n]]; If[Equal[s, 6], Print[n]], {n, 1, 1000}] (* Labos Elemer *)
Do[s=1+Divisors[n];s1=Flatten[Position[PrimeQ[s], True]]; (*analogous [suitably modified] pairs of programs yield A051225-A051230*) s2=Part[s, s1];If[Equal[s2, {2, 3}], Print[n]], {n, 1, 100}] (* Labos Elemer *)
Select[Range[600],Denominator[BernoulliB[#]]==6&] (* Harvey P. Dale, Dec 08 2011 *)

for(n=1,10^3,if(denominator(bernfrac(n))==6,print1(n,", "))); \\ Joerg Arndt, Oct 28 2014

is(n)=if(n%2,return(0)); fordiv(n/2,d,if(isprime(2*d+1)&&d>1, return(0))); 1 \\ Charles R Greathouse IV, Oct 28 2014

Additional comments and references from Sam Wagstaff, Dec 20 2005

A084645 Hypotenuses for which there exists a unique integer-sided right triangle.

Original entry on oeis.org

5, 10, 13, 15, 17, 20, 26, 29, 30, 34, 35, 37, 39, 40, 41, 45, 51, 52, 53, 55, 58, 60, 61, 68, 70, 73, 74, 78, 80, 82, 87, 89, 90, 91, 95, 97, 101, 102, 104, 105, 106, 109, 110, 111, 113, 115, 116, 117, 119, 120, 122, 123, 135, 136, 137, 140, 143, 146, 148, 149

Offset: 1

Eric W. Weisstein, Jun 01 2003

nonn

Numbers whose square is uniquely decomposable into the sum of two nonzero squares: these are those numbers with exactly one prime divisor of the form 4k+1 with multiplicity one. - Jean-Christophe Hervé, Nov 11 2013

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Zak Seidov)
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A002144, A006339, A046080, A046109, A083025.

Cf. A004144 (0), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

r[a_] := {b, c} /. {ToRules[ Reduce[0 < b < c && a^2 == b^2 + c^2, {b, c}, Integers]]}; Select[ Range[150], Length[r[#]] == 1 &] (* Jean-François Alcover, Oct 22 2012 *)

is_a084645(n) = #qfbsolve(Qfb(1,0,1),n^2,3)==3 \\ Hugo Pfoertner, Sep 28 2024

Terms are obtained by the products A004144(k)*A002144(p) for k, p > 0, ordered by increasing values. - Jean-Christophe Hervé, Nov 12 2013

A046080(a(n)) = 1, A046109(a(n)) = 12. - Jean-Christophe Hervé, Dec 01 2013

A002330 Value of y in the solution to p = x^2 + y^2, x <= y, with prime p = A002313(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 5, 7, 6, 8, 8, 9, 10, 10, 8, 11, 10, 11, 13, 10, 12, 14, 15, 13, 15, 16, 13, 14, 16, 17, 13, 14, 16, 18, 17, 18, 17, 19, 20, 20, 15, 17, 20, 21, 19, 22, 20, 21, 19, 20, 24, 23, 24, 18, 19, 25, 22, 25, 23, 26, 26, 22, 27, 26, 20, 25, 22, 26, 28, 25

Offset: 1

N. J. A. Sloane

nonn

			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. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
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).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
John Brillhart, Note on representing a prime as a sum of two squares, Math. Comp. 26 (1972), pp. 1011-1013.
A. J. C. Cunningham, Quadratic Partitions, Hodgson, London, 1904. [Annotated scans of selected pages]
K. Matthews, Serret's algorithm Server.
J. Todd, A problem on arc tangent relations, Amer. Math. Monthly, 56 (1949), 517-528.
Eric Weisstein's World of Mathematics, Fermat's 4n Plus 1 Theorem.

Cf. A002331, A002313, A002144.

a := []; for x from 0 to 50 do for y from x to 50 do p := x^2+y^2; if isprime(p) then a := [op(a),[p,x,y]]; fi; od: od: writeto(trans); for i from 1 to 158 do lprint(a[i]); od: # then sort the triples in "trans"

Flatten[#, 1]&[Table[PowersRepresentations[Prime[k], 2, 2], {k, 1, 142}]][[All, 2]] (* Jean-François Alcover, Jul 05 2011 *)

f(p)=my(s=lift(sqrt(Mod(-1,p))),x=p,t);if(s>p/2,s=p-s); while(s^2>p, t=s;s=x%s;x=t);s
forprime(p=2,1e3,if(p%4-3,print1(f(p)", "))) \\ Charles R Greathouse IV, Apr 24 2012

do(p)=qfbsolve(Qfb(1,0,1),p)[1]
forprime(p=2,1e3,if(p%4-3,print1(do(p)", "))) \\ Charles R Greathouse IV, Sep 26 2013

print1(1); forprimestep(p=5,1e3,4, print1(", "qfbcornacchia(1,p)[1])) \\ Charles R Greathouse IV, Sep 15 2021

a(n) = A096029(n) + A096030(n) + 1, for n>1. - Lekraj Beedassy, Jul 21 2004

a(n+1) = Max(A002972(n), 2*A002973(n)). - Reinhard Zumkeller, Feb 16 2010

A005098 Numbers k such that 4k + 1 is prime.

Original entry on oeis.org

1, 3, 4, 7, 9, 10, 13, 15, 18, 22, 24, 25, 27, 28, 34, 37, 39, 43, 45, 48, 49, 57, 58, 60, 64, 67, 69, 70, 73, 78, 79, 84, 87, 88, 93, 97, 99, 100, 102, 105, 108, 112, 114, 115, 127, 130, 135, 139, 142, 144, 148, 150, 153, 154, 160, 163, 165, 168, 169, 175, 177, 183

Offset: 1

N. J. A. Sloane

Sum of i-th and j-th triangular numbers, where i=A096029(n), j=A096030(n); i.e., a(n) = A000217(A096029(n)) + A000217(A096030(n)). - Lekraj Beedassy, Jun 16 2004
For every k in the sequence, there is exactly 1 square number that can be subtracted to leave a pronic (A002378). E.g., 27 - 25 = 2, 99 - 9 = 90. - Jon Perry, Nov 06 2010
See A208295 for details concerning the preceding Jon Perry comment. - Wolfdieter Lang, Mar 29 2012
a(k) appears in the o.g.f. for floor(A002144(k)*j^2/4), j >= 0, for k >= 1: x*(a(k)*(1 + x^2) + b(k)*x)/((1 - x)^3*(1 + x)), together with b(k) = (A002144(k) + 1)/2 = A119681(k). - Wolfdieter Lang, Aug 07 2013

Zak Seidov, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Wilson's Theorem.

See A002144 for the actual primes.

Cf. A002972, A002973, A173331, A173330, A023212.

a005098 = (`div` 4) . (subtract 1) . a002144
-- Reinhard Zumkeller, Mar 17 2013

[k: k in [0..10000] | IsPrime(4*k+1)] // Vincenzo Librandi, Nov 18 2010

a := []; for k from 1 to 500 do if isprime(4*k+1) then a := [op(a), k]; fi; od: A005098 := k->a[k];

Select[Range[200], PrimeQ[4# + 1] &] (* Harvey P. Dale, Apr 20 2011 *)

is(k)=isprime(4*k+1) \\ Charles R Greathouse IV, Nov 20 2012

a(n) = (A002144(n)-1)/4.

More terms from Ray Chandler, Jun 26 2004

Edited by Charles R Greathouse IV, Mar 17 2010

A053176 Primes p such that 2p+1 is composite.

Original entry on oeis.org

7, 13, 17, 19, 31, 37, 43, 47, 59, 61, 67, 71, 73, 79, 97, 101, 103, 107, 109, 127, 137, 139, 149, 151, 157, 163, 167, 181, 193, 197, 199, 211, 223, 227, 229, 241, 257, 263, 269, 271, 277, 283, 307, 311, 313, 317, 331, 337, 347, 349, 353, 367, 373, 379, 383

Offset: 1

Enoch Haga, Feb 29 2000

Primes not in A005384 = non-Sophie Germain primes.
Also, numbers n such that odd part of A005277(n) is prime. Proof by John Renze, Sep 30 2004
Sequence gives primes p such that B(2p) has denominator 6, where B(2n) are the Bernoulli numbers. - Benoit Cloitre, Feb 06 2002
Sequence gives all n such that the equation phi(x)=2n has no solution. - Benoit Cloitre, Apr 07 2002
A010051(a(n))*(1-A156660(a(n))) = 1; subsequence of A138887. - Reinhard Zumkeller, Feb 18 2009
Mersenne prime exponents > 3 must be in the union of this sequence and (A002144). - Roderick MacPhee, Jan 12 2017

			17 is a term because 2*17 + 1 = 35 is composite.

T. D. Noe, Table of n, a(n) for n = 1..10000
Lambert A'Campo, Every 7-Dimensional Abelian Variety over the p-adic Numbers has a Reducible L-adic Galois Representation, arXiv:2006.06737 [math.NT], 2020.

Cf. A005384, A005385, A059452, A059453, A059454, A059455, A059456, A007700, A005602, A023272, A023302, A023330, A156543, A156542.

[p: p in PrimesUpTo(12200) | not IsPrime(2*p+1)]; // Vincenzo Librandi, Jun 18 2015

Select[Prime[Range[1000]], ! PrimeQ[2 # + 1] &] (* Vincenzo Librandi, Jun 18 2015 *)

list(lim)=select(p->!isprime(2*p+1),primes(primepi(lim))) \\ Charles R Greathouse IV, Jul 25 2011

a(n) ~ n log n. - Charles R Greathouse IV, Feb 20 2012

A055025 Norms of Gaussian primes.

Original entry on oeis.org

2, 5, 9, 13, 17, 29, 37, 41, 49, 53, 61, 73, 89, 97, 101, 109, 113, 121, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 361, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461, 509, 521, 529, 541, 557, 569

Offset: 1

N. J. A. Sloane, Jun 09 2000

This is the range of the norm a^2 + b^2 of Gaussian primes a + b i. A239621 lists for each norm value a(n) one of the Gaussian primes as a, b with a >= b >= 0. In A239397, any of these (a, b) is followed by (b, a), except for a = b = 1. - Wolfdieter Lang, Mar 24 2014, edited by M. F. Hasler, Mar 09 2018
From Jean-Christophe Hervé, May 01 2013: (Start)
The present sequence is related to the square lattice, and to its division in square sublattices. Let's say that an integer n divides a lattice if there exists a sublattice of index n. Example: 2, 4, 5 divide the square lattice. Then A001481 (norms of Gaussian integers) is the sequence of divisors of the square lattice. Say that n is a "prime divisor" if the index-n sublattice is not contained in any other sublattice except the original lattice itself. The present sequence gives the "prime divisors" of the square lattice.
Similarly, A055664 (Norms of Eisenstein-Jacobi primes) is the sequence of "prime divisors" of the hexagonal lattice. (End)
The sequence is formed of 2, the prime numbers of form 4k+1, and the square of other primes (of form 4k+3). These are the primitive elements of A001481. With 0 and 1, they are the numbers that are uniquely decomposable in the sum of two squares. - Jean-Christophe Hervé, Nov 17 2013

			There are 8 Gaussian primes of norm 5, +-1+-2i and +-2+-i, but only two inequivalent ones (2+-i). In A239621 2+i is listed as 2, 1.

R. K. Guy, Unsolved Problems in Number Theory, A16.
L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. V.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Eric Weisstein's World of Mathematics, Gaussian prime
Wikipedia, Table of Gaussian integer factorizations
Index entries for Gaussian integers and primes

Cf. A055026, A055027, A055028, A055029, A055664-A055666, A001481.

Cf. A239397, A239621 (Gaussian primes).

Union[(#*Conjugate[#] & )[ Select[Flatten[Table[a + b*I, {a, 0, 23}, {b, 0, 23}]], PrimeQ[#, GaussianIntegers -> True] & ]]][[1 ;; 55]] (* Jean-François Alcover, Apr 08 2011 *)
(* Or, from formula: *) maxNorm = 569; s1 = Select[Range[1, maxNorm, 4], PrimeQ]; s3 = Select[Range[3, Sqrt[maxNorm], 4], PrimeQ]^2; Union[{2}, s1, s3]  (* Jean-François Alcover, Dec 07 2012 *)

list(lim)=my(v=List()); if(lim>=2, listput(v,2)); forprime(p=3,sqrtint(lim\1), if(p%4==3, listput(v,p^2))); forprime(p=5,lim, if(p%4==1, listput(v,p))); Set(v) \\ Charles R Greathouse IV, Feb 06 2017

isA055025(n)=(isprime(n) && n%4<3) || (issquare(n, &n) && isprime(n) && n%4==3) \\ Jianing Song, Aug 15 2023, based on Charles R Greathouse IV's program for A055664

Consists of 2; rational primes = 1 (mod 4) [A002144]; and squares of rational primes = 3 (mod 4) [A002145^2].

a(n) ~ 2n log n. - Charles R Greathouse IV, Feb 06 2017

More terms from Larry Reeves (larryr(AT)acm.org), Oct 03 2000

A003658 Fundamental discriminants of real quadratic fields; indices of primitive positive Dirichlet L-series.

Original entry on oeis.org

1, 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40, 41, 44, 53, 56, 57, 60, 61, 65, 69, 73, 76, 77, 85, 88, 89, 92, 93, 97, 101, 104, 105, 109, 113, 120, 124, 129, 133, 136, 137, 140, 141, 145, 149, 152, 156, 157, 161, 165, 168, 172, 173, 177, 181, 184, 185, 188, 193, 197

Offset: 1

N. J. A. Sloane, Mira Bernstein, Eric W. Weisstein

All the prime numbers in the set of positive fundamental discriminants are Pythagorean primes (A002144). - Paul Muljadi, Mar 28 2008

Record numbers of prime divisors (with multiplicity) are 1, 5, and 4*A002110(n) for n > 0. - Charles R Greathouse IV, Jan 21 2022

Henri Cohen, A Course in Computational Algebraic Number Theory, Springer, 1993, pp. 515-519.
M. Pohst and Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, 1989, page 432.
Paulo Ribenboim, Algebraic Numbers, Wiley, NY, 1972, p. 97.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..3001 from T. D. Noe)
Steven R. Finch, Class number theory, 2005. [Cached copy, with permission of the author]
Britta Habdank-Eichelsbacher, Unimodulare Gitter über Reell-Quadratischen Zahlkörpern, Ergänzungsreihe 95-005, Univ. Bielefeld, 1995. See Section 4.2.
Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
Eric Weisstein's World of Mathematics, Dirichlet L-Series.
Eric Weisstein's World of Mathematics, Fundamental Discriminant.
Eric Weisstein's World of Mathematics, Class Number.

Cf. A003652, A003657, A002144, A003646 (class numbers), A014000, A014046, A086669, A232931, A290098.

Union of A039955 and 4*A230375.

fundamentalDiscriminantQ[d_] := Module[{m, mod = Mod[d, 4]}, If[mod > 1, Return[False]]; If[mod == 1, Return[SquareFreeQ[d] && d != 1]]; m = d/4; Return[SquareFreeQ[m] && Mod[m, 4] > 1]; ]; Join[{1}, Select[Range[200], fundamentalDiscriminantQ]] (* Jean-François Alcover, Nov 02 2011, after Eric W. Weisstein *)
Select[Range[200], NumberFieldDiscriminant@Sqrt[#] == # &]  (* Alonso del Arte, Apr 02 2014, based on Arkadiusz Wesolowski's program for A094612 *)
max = 200; Drop[Select[Union[Table[Abs[MoebiusMu[n]] * n * 4^Boole[Not[Mod[n, 4] == 1]], {n, max}]], # < max &], 1] (* Alonso del Arte, Apr 02 2014 *)

v=[]; for(n=1,500,if(isfundamental(n),v=concat(v,n))); v

list(lim)=my(v=List()); forsquarefree(n=1,lim\4, listput(v, if(n[1]%4==1, n[1], 4*n[1]))); forsquarefree(n=lim\4+1, lim\1, if(n[1]%4==1, listput(v,n[1]))); Set(v) \\ Charles R Greathouse IV, Jan 21 2022

def is_fundamental(d):
    r = d % 4
    if r > 1 : return False
    if r == 1: return (d != 1) and is_squarefree(d)
    q = d // 4
    return is_squarefree(q) and (q % 4 > 1)
[1] + [n for n in (1..200) if is_fundamental(n)] # Peter Luschny, Oct 15 2018

Squarefree numbers (multiplied by 4 if not == 1 (mod 4)).

a(n) ~ (Pi^2/3)*n. There are (3/Pi^2)*x + O(sqrt(x)) terms up to x. - Charles R Greathouse IV, Jan 21 2022

More terms from Eric W. Weisstein and Jason Earls, Jun 19 2001

cp's OEIS Frontend

A014442 Largest prime factor of n^2 + 1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

PARI

Formula

A027862 Primes of the form j^2 + (j+1)^2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

Extensions

A004613 Numbers that are divisible only by primes congruent to 1 mod 4.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

Formula

A051222 Numbers k such that Bernoulli number B_{k} has denominator 6.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A084645 Hypotenuses for which there exists a unique integer-sided right triangle.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A002330 Value of y in the solution to p = x^2 + y^2, x <= y, with prime p = A002313(n).

Views

Author

Keywords

Examples

References

Links