A002970 -id:A002970 - cp's OEIS frontend

A001912 Numbers k such that 4*k^2 + 1 is prime.

1, 2, 3, 5, 7, 8, 10, 12, 13, 18, 20, 27, 28, 33, 37, 42, 45, 47, 55, 58, 60, 62, 63, 65, 67, 73, 75, 78, 80, 85, 88, 90, 92, 102, 103, 105, 112, 115, 118, 120, 125, 128, 130, 132, 135, 140, 142, 150, 153, 157, 163, 170, 175, 192, 193, 198, 200

Offset: 1

N. J. A. Sloane

Complement of A094550. - Hermann Stamm-Wilbrandt, Sep 16 2014
Positive integers whose square is the sum of two triangular numbers in exactly one way (A000217(k) + A000217(k+1) = k*(k+1)/2 + (k+1)*(k+2)/2 = (k+1)^2). In other words, positive integers k such that A052343(k^2) = 1. - Altug Alkan, Jul 06 2016
4*a(n)^2 + 1 = A002496(n+1). - Hal M. Switkay, Apr 03 2022

E. Kogbetliantz and A. Krikorian, Handbook of First Complex Prime Numbers, Gordon and Breach, NY, 1971, p. 1.
M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 11.
C. S. Ogilvy, Tomorrow's Math. 2nd ed., Oxford Univ. Press, 1972, p. 116.
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).

T. D. Noe, Table of n, a(n) for n = 1..10000
A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]
E. Kogbetliantz and A. Krikorian Handbook of First Complex Prime Numbers, Gordon and Breach, NY, 1971 [Annotated scans of a few pages]
Marek Wolf, Search for primes of the form m^2+1, arXiv:0803.1456 [math.NT], 2008-2010.

Cf. A002496, A005574, A062325, A090693, A094550, A214517 (first differences).

[n: n in [1..100] | IsPrime(4*n^2+1)] // Vincenzo Librandi, Nov 21 2010

A001912 := proc(n)
    option remember;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isprime(4*a^2+1) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Aug 09 2012

Select[Range[200], PrimeQ[4#^2 + 1] &] (* Alonso del Arte, Dec 20 2013 *)

is(n)=isprime(4*n^2 + 1) \\ Charles R Greathouse IV, Apr 28 2015

a(n) = A005574(n+1)/2.

A114272 Numbers k such that k^2 + 9 is prime.

Original entry on oeis.org

2, 8, 10, 20, 32, 38, 40, 52, 58, 62, 70, 82, 88, 98, 100, 110, 112, 118, 140, 142, 160, 170, 188, 190, 200, 202, 212, 218, 220, 242, 298, 308, 320, 332, 350, 358, 368, 380, 382, 400, 410, 412, 422, 448, 472, 482, 490, 502, 512, 530, 538, 542, 568, 572, 578

Offset: 1

Zak Seidov, Nov 19 2005

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Other sequences of the type "Numbers k such that k^2 + i is prime": A005574 (i=1), A067201 (i=2), A049422 (i=3), A007591 (i=4), A078402 (i=5), A114269 (i=6), A114270 (i=7), A114271 (i=8), this sequence (i=9), A114273 (i=10), A114274 (i=11), A114275 (i=12).

lst={};Do[If[PrimeQ[n^2+9], AppendTo[lst, n]], {n, 600}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 26 2008 *)
Select[Range[2,600,2],PrimeQ[#^2+9]&] (* Harvey P. Dale, Feb 28 2023 *)

is(n)=isprime(n^2+9) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = 2 * A002970(n). - Michel Marcus, Jan 20 2015

A002972 a(n) is the odd member of {x,y}, where x^2 + y^2 is the n-th prime of the form 4i+1.

Original entry on oeis.org

1, 3, 1, 5, 1, 5, 7, 5, 3, 5, 9, 1, 3, 7, 11, 7, 11, 13, 9, 7, 1, 15, 13, 15, 1, 13, 9, 5, 17, 13, 11, 9, 5, 17, 7, 17, 19, 1, 3, 15, 17, 7, 21, 19, 5, 11, 21, 19, 13, 1, 23, 5, 17, 19, 25, 13, 25, 23, 1, 5, 15, 27, 9, 19, 25, 17, 11, 5, 25, 27, 23, 29, 29, 25, 23, 19, 29, 13, 31, 31

Offset: 1

N. J. A. Sloane

nonn

It appears that the terms in this sequence are the absolute values of the terms in A046730. - Gerry Myerson, Dec 02 2010

"the n-th prime of the form 4i+1" is A005098(n). - Rainer Rosenthal, Aug 24 2022

			The 2nd prime of the form 4i+1 is 13 = 2^2 + 3^2, so a(2)=3.

E. Kogbetliantz and A. Krikorian, Handbook of First Complex Prime Numbers, Gordon and Breach, NY, 1971, p. 243.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Rainer Rosenthal, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
E. Kogbetliantz and A. Krikorian, Handbook of First Complex Prime Numbers, Gordon and Breach, NY, 1971. [Annotated scans of a few pages]
Stan Wagon, Editor's Corner: The Euclidean Algorithm Strikes Again, The American Mathematical Monthly, vol. 97, no. 2, 1990, pp. 125-29. [Description of efficient decomposition algorithm implemented in PARI program]

Cf. A002144, A002973, A005098, A261858.

pmax = 1000; odd[p_] := Module[{k, m}, 2m+1 /. ToRules[Reduce[k>0 && m >= 0 && (2k)^2 + (2m+1)^2 == p, {k, m}, Integers]]]; For[n=1; p=5, pJean-François Alcover, Feb 26 2016 *)

decomp2sq(p) = {my (m=(p-1)/4, r, x, limit=ceil(sqrt(p))); if (p>4 && denominator(m)==1, forprime (c=2,oo, if (!issquare(Mod(c,p)), r=c; break)); x=lift (Mod(r,p)^m); until (px%2,decomp2sq(p))[1],", "))) \\ Hugo Pfoertner, Aug 27 2022

a(n) = Min(A173330(n), A002144(n) - A173330(n)). - Reinhard Zumkeller, Feb 16 2010
a(n)^2 + 4*A002973(n)^2 = A002144(n); A002331(n+1) = Min(a(n),2*A002973(n)) and A002330(n+1) = Max(a(n),2*A002973(n)). - Reinhard Zumkeller, Feb 16 2010
(a(n) - 1)/2 = A208295(n), n >= 1. - Wolfdieter Lang, Mar 03 2012
a(A267858(k)) == 1 (mod 4), k >= 1. - Wolfdieter Lang, Feb 18 2016

Better description from Jud McCranie, Mar 05 2003

A002973 a(n) is half of the even member of {x,y}, where x^2 + y^2 is the n-th prime of the form 4i+1.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 3, 4, 4, 2, 5, 5, 4, 2, 5, 3, 1, 5, 6, 7, 1, 4, 2, 8, 5, 7, 8, 1, 6, 7, 8, 9, 4, 9, 5, 3, 10, 10, 7, 6, 10, 2, 5, 11, 10, 5, 7, 10, 12, 4, 12, 9, 8, 2, 11, 3, 6, 13, 13, 11, 1, 13, 10, 6, 11, 13, 14, 7, 5, 9, 2, 3, 8, 10, 12, 5, 14, 2, 3, 14, 11, 15, 16, 16, 5, 15, 1, 8, 11

Offset: 1

N. J. A. Sloane

nonn

a(n) is odd iff x^2 + y^2 == 5 (mod 8). [Vladimir Shevelev, Jul 12 2009]

A002972(n)^2 + 4*a(n)^2 = A002144(n); A002331(n+1) = Min(A002972(n),2*a(n)) and A002330(n+1) = Max(A002972(n),2*a(n)). [Reinhard Zumkeller, Feb 16 2010]

			The 3rd prime of the form 4i+1 is 17 = 1^2 + 4^2, so a(3) = 4/2 = 2.

E. Kogbetliantz and A. Krikorian, Handbook of First Complex Prime Numbers, Gordon and Breach, NY, 1971, p. 243.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Rainer Rosenthal, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
E. Kogbetliantz and A. Krikorian, Handbook of First Complex Prime Numbers, Gordon and Breach, NY, 1971. [Annotated scans of a few pages]

Cf. A002144, A002972, A005098.

pmax = 1000; k[p_] := Module[{k, m}, k /. ToRules[Reduce[k>0 && m >= 0 && (2k)^2 + (2m+1)^2 == p, {k, m}, Integers]]]; For[n=1; p=5, pJean-François Alcover, Feb 26 2016 *)

\\ use function decomp2sq from A002972
forprime (p=5, 1000, if (p%4==1, print1(select(x->!(x%2),decomp2sq(p))[1]/2,", "))) \\ Hugo Pfoertner, Aug 27 2022

a(n) = Min(A173331(n), A002144(n) - A173331(n)) / 2. [Reinhard Zumkeller, Feb 16 2010]

Better description from Jud McCranie, Mar 05 2003

A002971 Numbers k such that 4*k^2 + 25 is prime.

Original entry on oeis.org

1, 2, 3, 4, 8, 9, 11, 12, 13, 14, 16, 17, 18, 21, 23, 26, 29, 34, 36, 37, 38, 47, 48, 49, 51, 53, 54, 56, 62, 63, 66, 67, 68, 69, 73, 74, 77, 79, 82, 83, 91, 99, 101, 102, 103, 107, 108, 114, 116, 118, 122, 131, 134, 141, 142, 147, 148, 151, 154, 156, 157, 158, 159, 164

Offset: 1

N. J. A. Sloane

E. Kogbetliantz and A. Krikorian, Handbook of First Complex Prime Numbers, Gordon and Breach, NY, 1971, p. 1.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
E. Kogbetliantz and A. Krikorian Handbook of First Complex Prime Numbers, Gordon and Breach, NY, 1971 [Annotated scans of a few pages]

[n: n in [0..600] |IsPrime(4*n^2+25)]; // Vincenzo Librandi, Nov 21 2010

Select[Range[200], PrimeQ[4 #^2 + 25] &] (* Vincenzo Librandi, Mar 19 2014 *)

is(n)=isprime(4*n^2+25) \\ Charles R Greathouse IV, Jun 06 2017

cp's OEIS Frontend

A001912 Numbers k such that 4*k^2 + 1 is prime.

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A114272 Numbers k such that k^2 + 9 is prime.

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A002972 a(n) is the odd member of {x,y}, where x^2 + y^2 is the n-th prime of the form 4i+1.

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A002973 a(n) is half of the even member of {x,y}, where x^2 + y^2 is the n-th prime of the form 4i+1.

Views

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Keywords

Comments

Examples

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Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A002971 Numbers k such that 4*k^2 + 25 is prime.

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References

Links

Programs

Magma

Mathematica

PARI