A005098 -id:A005098 - cp's OEIS frontend

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.

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

A123998 Numbers k such that 2k+1 and 4k+1 are primes.

Original entry on oeis.org

1, 3, 9, 15, 18, 39, 48, 69, 78, 99, 105, 114, 135, 153, 165, 168, 183, 189, 219, 249, 273, 288, 300, 303, 309, 330, 345, 363, 405, 414, 438, 468, 483, 498, 504, 534, 585, 618, 639, 648, 699, 714, 729, 765, 804, 813, 828, 879, 933, 1005, 1014, 1044, 1065, 1068

Offset: 1

Artur Jasinski, Oct 31 2006

Note that if n == 1 (mod 3) then 2n+1 is not prime (except n=1); and if n == 2 (mod 3) then 4n+1 is not prime. Therefore n must be a multiple of 3, except for n=1. - Max Alekseyev, Nov 02 2006

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
J. O'Rourke, Why are this operator's primes the Sophie Germain primes?

Cf. A005097, A005098, A124408, A124409, A124410, A124411, A071576.

[n: n in [0..1100] |IsPrime(2*n+1) and IsPrime(4*n+1)]; // Vincenzo Librandi, Apr 17 2013

Select[Range[1100], And @@ PrimeQ /@ ({2, 4}*# + 1) &] (* Ray Chandler, Nov 20 2006 *)

is(k) = isprime(2*k+1) && isprime(4*k+1); \\ Jinyuan Wang, Aug 04 2019

Extended by Ray Chandler, Nov 20 2006

A123986 Numbers n for which 4n+1 and 4n+3 are primes.

Original entry on oeis.org

1, 4, 7, 10, 25, 34, 37, 49, 67, 70, 115, 130, 142, 154, 160, 202, 205, 214, 220, 262, 265, 307, 319, 322, 325, 370, 424, 430, 469, 487, 499, 520, 532, 535, 559, 577, 595, 637, 664, 682, 697, 700, 742, 814, 832, 847, 865, 889, 895, 955, 979, 982, 1000, 1012, 1039

Offset: 1

Artur Jasinski, Oct 30 2006

nonn

All terms == 1 mod 3. - Zak Seidov, Dec 02 2011

Intersection of A005098 and A095278. - Michel Marcus, Jan 31 2015

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A005098, A005099, A095278.

[n: n in [0..1100] |IsPrime(4*n+1) and IsPrime(4*n+3)]; // Vincenzo Librandi, Feb 01 2015

Select[Range[1100], And @@ PrimeQ /@ ({1, 3} + 4#) &] (* Ray Chandler, Nov 05 2006 *)
nn=10000;k=0;x=1;re=Reap[While[kZak Seidov, Dec 02 2011 *)

Extended by Ray Chandler, Nov 05 2006

A111199 Numbers k such that 4k + 9 is prime.

Original entry on oeis.org

1, 2, 5, 7, 8, 11, 13, 16, 20, 22, 23, 25, 26, 32, 35, 37, 41, 43, 46, 47, 55, 56, 58, 62, 65, 67, 68, 71, 76, 77, 82, 85, 86, 91, 95, 97, 98, 100, 103, 106, 110, 112, 113, 125, 128, 133, 137, 140, 142, 146, 148, 151, 152, 158, 161, 163, 166, 167, 173, 175, 181, 187

Offset: 1

Parthasarathy Nambi, Oct 24 2005

nonn

			For k=98, 4*k + 9 = 401 (prime).

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A005098, A095278, A089986, A153238, A144562.

is(n)=isprime(4*n+9) \\ Charles R Greathouse IV, Feb 20 2017

a(n) = A005098(n+1) - 2. - R. J. Mathar, Sep 23 2009

More terms from R. J. Mathar, Sep 23 2009

A171555 Numbers of the form prime(n)*(prime(n)-1)/4.

Original entry on oeis.org

5, 39, 68, 203, 333, 410, 689, 915, 1314, 1958, 2328, 2525, 2943, 3164, 4658, 5513, 6123, 7439, 8145, 9264, 9653, 13053, 13514, 14460, 16448, 18023, 19113, 19670, 21389, 24414, 25043, 28308, 30363, 31064, 34689, 37733, 39303, 40100, 41718, 44205, 46764, 50288

Offset: 1

Juri-Stepan Gerasimov, Dec 11 2009

nonn

The halves of even numbers of the form p(p-1)/2 for p prime.

Sum of the quadratic residues of primes of the form 4k + 1. For example, a(3)=68 because 17 is the 3rd prime of the form 4k + 1 and the quadratic residues of 17 are 1, 4, 9, 16, 8, 2, 15, 13 which sum to 68. This sum is also the sum of the quadratic nonresidues. Cf. A230077. - Geoffrey Critzer, May 07 2015

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see Exercise 2.21 p. 110.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Aebi, Christian, and Grant Cairns. Sums of Quadratic residues and nonresidues, arXiv preprint arXiv:1512.00896 (2015).

Cf. A005098, A007742, A008837.

Sums of residues, nonresidues, and their differences, for p == 1 (mod 4), p == 3 (mod 4), and all p: A171555; A282035, A282036, A282037; A076409, A125615, A282038.

Table[Table[Mod[a^2, p], {a, 1, (p - 1)/2}] // Total, {p,
Select[Prime[Range[100]], Mod[#, 4] == 1 &]}] (* Geoffrey Critzer, May 07 2015 *)
Select[(# (#-1))/4&/@Prime[Range[100]],IntegerQ] (* Harvey P. Dale, Dec 24 2022 *)

lista(nn) = forprime(p=2, nn, if ((p % 4)==1, print1(p*(p-1)/4, ", "))); \\ Michel Marcus, Mar 23 2016

Corrected (16448 inserted, 25043 inserted) by R. J. Mathar, May 22 2010

A124408 Numbers k such that 2k+1, 4k+1 and 6k+1 are primes.

Original entry on oeis.org

1, 3, 18, 105, 135, 153, 165, 168, 300, 363, 585, 618, 648, 765, 828, 1110, 1140, 1278, 1518, 1530, 1533, 2130, 2223, 2400, 2475, 2613, 2790, 2925, 3075, 3180, 3345, 3420, 3483, 3810, 3840, 3843, 3933, 4008, 4083, 4095, 4143, 4260, 4263, 4323, 4470, 4545

Offset: 1

Artur Jasinski, Oct 31 2006

nonn

Jinyuan Wang, Table of n, a(n) for n = 1..10000

Cf. A005097, A005098, A024899, A123998, A124409, A124410, A124411, A071576.

Select[Range[4600], And @@ PrimeQ /@ ({2, 4, 6}*# + 1) &] (* Ray Chandler, Nov 20 2006 *)

is(k) = sum(j = 1, 3, isprime(2*j*k+1)) == 3; \\ Jinyuan Wang, Aug 04 2019

A124409 Numbers k such that 2k+1, 4k+1, 6k+1 and 8k+1 are primes.

Original entry on oeis.org

165, 765, 1530, 2130, 2475, 3420, 5415, 7695, 9060, 11505, 12705, 13020, 15885, 16650, 20055, 20745, 22530, 24915, 26940, 29670, 32925, 35070, 36885, 39270, 44370, 47730, 48465, 54735, 55860, 56310, 58860, 65655, 66600, 67365, 67650

Offset: 1

Artur Jasinski, Oct 31 2006

nonn

Jinyuan Wang, Table of n, a(n) for n = 1..2000

Cf. A005097, A005098, A024899, A005123, A123998, A124408, A124410, A124411, A071576.

Select[Range[68000], And @@ PrimeQ /@ ({2, 4, 6, 8}*# + 1) &] (* Ray Chandler, Nov 20 2006 *)

is(k) = sum(j = 1, 4, isprime(2*j*k+1)) == 4; \\ Jinyuan Wang, Aug 04 2019

A124410 Numbers k such that 2k+1, 4k+1, 6k+1, 8k+1 and 10k+1 are primes.

Original entry on oeis.org

5415, 12705, 13020, 44370, 82950, 98280, 105525, 112200, 115140, 123855, 134250, 134460, 187740, 188745, 210165, 225705, 247170, 256410, 296310, 302085, 367875, 375645, 382890, 399585, 404040, 476340, 487830, 526845, 532095, 566430, 578085

Offset: 1

Artur Jasinski, Oct 31 2006

nonn

Jinyuan Wang, Table of n, a(n) for n = 1..1000

Cf. A005097, A005098, A024899, A005123, A024912, A123998, A124408, A124409, A124411, A071576.

Select[Range[600000], And @@ PrimeQ /@ ({2, 4, 6, 8, 10}*# + 1) &] (* Ray Chandler, Nov 20 2006 *)

is(k) = sum(j = 1, 5, isprime(2*j*k+1)) == 5; \\ Jinyuan Wang, Aug 04 2019

A124411 Numbers k such that 2k+1, 4k+1, 6k+1, 8k+1, 10k+1 and 12k+1 are primes.

Original entry on oeis.org

12705, 13020, 105525, 256410, 966840, 1707510, 1944495, 2310000, 2478630, 3132675, 3836070, 3976770, 4112430, 4532325, 5499585, 5920005, 6610485, 7390845, 8552250, 10739505, 11120340, 12231450, 12338130, 13243230, 16467255

Offset: 1

Artur Jasinski, Oct 31 2006

nonn

Jinyuan Wang, Table of n, a(n) for n = 1..100

Cf. A005097, A005098, A024899, A005123, A024912, A110801, A123998, A124408, A124409, A124410, A071576.

Select[Range[10^7], And @@ PrimeQ /@ ({2, 4, 6, 8, 10, 12}*# + 1) &] (* Ray Chandler, Nov 20 2006 *)

is(k) = sum(j = 1, 6, isprime(2*j*k+1)) == 6; \\ Jinyuan Wang, Aug 04 2019

Extended by Ray Chandler, Nov 20 2006

cp's OEIS Frontend

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.

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A123998 Numbers k such that 2k+1 and 4k+1 are primes.

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Magma

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A123986 Numbers n for which 4n+1 and 4n+3 are primes.

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Magma

Mathematica

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A111199 Numbers k such that 4k + 9 is prime.

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Programs

PARI

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A171555 Numbers of the form prime(n)*(prime(n)-1)/4.

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Mathematica

PARI

Extensions

A124408 Numbers k such that 2k+1, 4k+1 and 6k+1 are primes.

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