A005098 -id:A005098 - cp's OEIS frontend

A089986 Numbers n such that 4n + 7 is prime.

-1, 0, 1, 3, 4, 6, 9, 10, 13, 15, 16, 18, 19, 24, 25, 30, 31, 33, 36, 39, 40, 43, 46, 48, 51, 54, 55, 58, 61, 64, 66, 69, 75, 76, 81, 85, 88, 90, 93, 94, 103, 106, 108, 109, 114, 115, 118, 120, 121, 123, 124, 129, 135, 139, 141, 145, 148, 150, 153, 156, 159, 160, 163, 169

Offset: 1

Giovanni Teofilatto, Jan 13 2004

sign

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988.
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta,UTET, CittaStudiEdizioni, Milano 1997.

Cf. A005099 ((( Primes = -1 mod 4 ) + 1)/4), A005098 (4n+1 is prime), A095278 (4n+3 is prime), A111215 (4n+5 is prime).

[ n: n in [-1..170] | IsPrime(4*n+7) ];

A089986:=n->`if`(isprime(4*n+7), n, NULL): seq(A089986(n), n=-1..200); # Wesley Ivan Hurt, Sep 18 2014

Select[Range[-1, 200], PrimeQ[4 # + 7] &] (* Wesley Ivan Hurt, Sep 18 2014 *)

is(n)=isprime(4*n+7) \\ Charles R Greathouse IV, Feb 20 2017

a(n) = A005099(n) - 2 = A095278(n) - 1.

Edited and extended by Klaus Brockhaus, Dec 22 2008

A173331 Second of two intermediate sequences for integral solution of A002144(n)=x^2+y^2.

Original entry on oeis.org

2, 2, 13, 2, 31, 4, 2, 55, 8, 81, 4, 91, 99, 105, 133, 10, 6, 2, 10, 181, 183, 227, 8, 237, 16, 10, 14, 265, 2, 301, 303, 16, 18, 8, 355, 379, 6, 381, 389, 14, 421, 429, 453, 451, 487, 20, 531, 543, 20, 24, 585, 24, 18, 16, 637, 631, 655, 12, 651, 675, 22, 731, 26, 741, 757

Offset: 1

Reinhard Zumkeller, Feb 16 2010

nonn

a(n) = A173330(n)*A010050(A005098(n)) mod A002144(n);

A002973(n) = MIN(a(n), A002144(n) - a(n)) / 2.

			n=7: A002144(7) = 53 = 4*13 + 1,
a(7) = A173330(7) * 26! mod 53 = 7*403291461126605635584000000 mod 53 = 2,
A002973(7) = MIN(2, 53 - 2) / 2 = 1;
n=8: A002144(8) = 61 = 4*15 + 1,
a(8) = A173330(8) * 30! mod 61 = 5*265252859812191058636308480000000 mod 61 = 55,
A002973(8) = MIN(55, 61 - 55) / 2 = 3.

H. Davenport, The Higher Arithmetic (Cambridge University Press 7th ed., 1999), ch. V.3, p.122.

Cf. A123072, A000142.

a(n) = ((2k)! / 2(k!))^2 mod p, where p = 4*k+1 = A002144(n).

A096029 Values (x+y-1)/2, where x^2+y^2=p runs over the Pythagorean primes A002144.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 5, 6, 7, 7, 8, 8, 7, 9, 9, 7, 8, 10, 9, 8, 11, 11, 10, 9, 12, 12, 12, 11, 12, 12, 13, 12, 10, 11, 14, 14, 13, 12, 14, 13, 15, 15, 16, 16, 12, 15, 14, 17, 17, 14, 17, 15, 17, 13, 15, 18, 14, 17, 19, 18, 19, 18, 16, 19, 18, 20, 16, 17, 20, 21, 21, 19, 20

Offset: 1

Lekraj Beedassy, Jun 16 2004

nonn

Cf. A005098, A096030.

a(n)=(A079886(n) - 1)/2

More terms from Ray Chandler, Jun 26 2004

A096030 Values (y-x-1)/2, where x^2+y^2=p,(xA002144.

Original entry on oeis.org

0, 0, 1, 1, 2, 0, 2, 0, 2, 1, 2, 4, 3, 0, 3, 1, 2, 5, 0, 2, 6, 6, 2, 5, 7, 1, 2, 5, 7, 0, 1, 3, 6, 4, 5, 3, 6, 9, 8, 0, 2, 6, 8, 4, 8, 4, 5, 2, 3, 11, 7, 9, 0, 1, 10, 4, 9, 5, 12, 10, 3, 12, 8, 0, 6, 2, 7, 11, 5, 8, 2, 12, 11, 4, 1, 2, 9, 7, 13, 12, 6, 0, 9, 14, 13, 10, 8, 15, 6, 1, 2, 3, 12, 14, 9, 0, 2

Offset: 1

Lekraj Beedassy, Jun 16 2004

nonn

Cf. A005098, A096029.

a(n)=(A079887(n) - 1)/2.

More terms from Ray Chandler, Jun 26 2004

A111215 Numbers k such that 4k + 5 is prime.

Original entry on oeis.org

0, 2, 3, 6, 8, 9, 12, 14, 17, 21, 23, 24, 26, 27, 33, 36, 38, 42, 44, 47, 48, 56, 57, 59, 63, 66, 68, 69, 72, 77, 78, 83, 86, 87, 92, 96, 98, 99, 101, 104, 107, 111, 113, 114, 126, 129, 134, 138, 141, 143, 147, 149, 152, 153, 159, 162, 164, 167, 168, 174, 176, 182, 188

Offset: 1

Parthasarathy Nambi, Oct 24 2005

			If k=99 then 4k + 5 = 401 (prime).

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

Cf. A005098, A095278.

[n: n in [0..200] | IsPrime(4*n+5)]; // Vincenzo Librandi, Nov 13 2010

Select[Range[0,200],PrimeQ[4#+5]&] (* Harvey P. Dale, Dec 07 2013 *)

is(n)=isprime(4*n+5) \\ Charles R Greathouse IV, Aug 31 2015

a(n) = A005098(n) - 1. - Daniel Starodubtsev, Feb 10 2020

A173330 First of two intermediate sequences for integral solution of A002144(n)=x^2+y^2.

Original entry on oeis.org

1, 10, 1, 5, 1, 5, 46, 5, 70, 5, 9, 1, 106, 106, 126, 142, 146, 13, 9, 186, 1, 214, 13, 226, 1, 13, 9, 5, 17, 13, 306, 9, 5, 17, 366, 17, 378, 1, 406, 406, 17, 442, 21, 442, 5, 510, 21, 538, 13, 1, 570, 5, 17, 598, 25, 13, 25, 650, 1, 5, 694, 706, 9, 742, 25, 17, 786, 5, 25

Offset: 1

Reinhard Zumkeller, Feb 16 2010

nonn

A002972(n) = MIN(a(n), A002144(n) - a(n)).

			n=7: A002144(7) = 53 = 4*13 + 1,
a(7) = 26! / (2*(13!)^2) mod 53 = 403291461126605635584000000/77551576087265280000 mod 53 = 5200300 mod 53 = 46,
A002972(7) = MIN(46, 53 - 46) = 7;
n=8: A002144(8) = 61 = 4*15 + 1,
a(8) = 30! / (2*(15!)^2) mod 61 = 265252859812191058636308480000000/3420024505448398848000000 mod 61 = 77558760 mod 61 = 5,
A002972(8) = MIN(5, 61 - 5) = 5.

H. Davenport, The Higher Arithmetic (Cambridge University Press 7th ed., 1999), ch. V.3, p.122.

Cf. A173331, A001700, A010050, A000142, A005098.

a(n) = (2k)! / 2(k!)^2 mod p, where p = 4*k+1 = A002144(n).

A094896 If 4n+1 is prime and 4n+3 is not prime then a(n)=4*n+1, else a(n)=0.

Original entry on oeis.org

0, 0, 0, 13, 0, 0, 0, 0, 0, 37, 0, 0, 0, 53, 0, 61, 0, 0, 73, 0, 0, 0, 89, 0, 97, 0, 0, 109, 113, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 157, 0, 0, 0, 173, 0, 181, 0, 0, 193, 0, 0, 0, 0, 0, 0, 0, 0, 229, 233, 0, 241, 0, 0, 0, 257, 0, 0, 0, 0, 277, 0, 0, 0, 293, 0, 0, 0, 0, 313, 317, 0, 0, 0, 0, 337, 0, 0

Offset: 0

Roger L. Bagula, Jun 14 2004

nonn

Cf. A005098, A095277, A094897.

[IsPrime(4*n+1) and not IsPrime(4*n+3) select 4*n+1 else 0:n in [0..86]]; // Marius A. Burtea, Nov 15 2019

A094896 := proc(n)
    if isprime(4*n+1) and not isprime(4*n+3) then
        4*n+1;
    else
        0;
    end if;
end proc:
seq(A094896(n),n=0..86) ; # R. J. Mathar, Nov 15 2019

a=Table[If[PrimeQ[4*n+1]==True&&PrimeQ[4*n+3]==False, 4*n+1, 0], {n, 0, 200}]

A119681 Odd numbers n such that 2n-1 is prime.

Original entry on oeis.org

3, 7, 9, 15, 19, 21, 27, 31, 37, 45, 49, 51, 55, 57, 69, 75, 79, 87, 91, 97, 99, 115, 117, 121, 129, 135, 139, 141, 147, 157, 159, 169, 175, 177, 187, 195, 199, 201, 205, 211, 217, 225, 229, 231, 255, 261, 271, 279, 285, 289, 297, 301, 307, 309, 321, 327, 331

Offset: 1

Roger L. Bagula, Jun 12 2006

a(k) appears in the o.g.f. for floor(A002144(k)*n^2/4), for k >=1: x*(b(k)*(1 + x^2) + a(k)*x)/((1 - x)^3*(1 + x)), together with b(k) = (A002144(k)-1)/4 = A005098(k). - Wolfdieter Lang, Aug 07 2013

Select[Range[1,331,2],PrimeQ[2#-1]&] (* Harvey P. Dale, Feb 20 2013 *)

is(n)=n%2 && isprime(2*n-1) \\ Charles R Greathouse IV, Jun 06 2017

a(n) = (A002144(n) + 1)/2, n >= 1. - Wolfdieter Lang, Aug 07 2013

Edited by N. J. A. Sloane, Sep 07 2006

A163366 a(n) = (-1)^floor((prime(n)+2)/2) mod prime(n).

Original entry on oeis.org

1, 1, 4, 1, 1, 12, 16, 1, 1, 28, 1, 36, 40, 1, 1, 52, 1, 60, 1, 1, 72, 1, 1, 88, 96, 100, 1, 1, 108, 112, 1, 1, 136, 1, 148, 1, 156, 1, 1, 172, 1, 180, 1, 192, 196, 1, 1, 1, 1, 228, 232, 1, 240, 1, 256, 1, 268, 1, 276, 280, 1, 292, 1, 1, 312, 316, 1, 336, 1, 348, 352, 1, 1, 372, 1

Offset: 1

Peter Luschny, Jul 25 2009

nonn

Remove the '1's from the sequence to get A152680.

Product modulo p of the quadratic residues of p, where p = prime(n). [Jonathan Sondow, May 14 2010]

			a(4) = 1 because the quadratic residues of prime(4) = 7 are 1, 2, and 4, and 1*2*4 = 8 == 1 (mod 7). - _Jonathan Sondow_, May 14 2010

Carl-Erik Froeberg, On sums and products of quadratic residues, BIT, Nord. Tidskr. Inf.-behandl. 11 (1971) 389-398. [Jonathan Sondow, May 14 2010]

G. C. Greubel, Table of n, a(n) for n = 1..1000
Rahul Gupta, Algorithmic Number Theory, Section 24.5 [_Jonathan Sondow_, May 14 2010]

Cf. A152680, A005098, A002144, A009003.

Cf. A177860, A177863. - Jonathan Sondow, May 14 2010

seq((-1)^iquo(ithprime(i)+2,2) mod ithprime(i),i=1..113);

Table[Mod[ Apply[Times, Flatten[Position[ Table[JacobiSymbol[i, Prime[n]], {i, 1, Prime[n] - 1}], 1]]], Prime[n]], {n, 1, 80}] (* Jonathan Sondow, May 14 2010 *)

a(n)*A177863(n) == -1 (mod prime(n)), by Wilson's theorem. - Jonathan Sondow, May 14 2010

a(n) = A177860(n) modulo prime(n). - Jonathan Sondow, May 14 2010

A055131 Those composite s for which A055095[s] = 2.

Original entry on oeis.org

15, 39, 51, 87, 111, 123, 159, 183, 219, 267, 291, 303, 327, 339, 411, 447, 471, 519, 543, 579, 591, 687, 699, 723, 771, 807, 831, 843, 879, 939, 951, 1011, 1047, 1059, 1119, 1167, 1191, 1203, 1227, 1263, 1299, 1347, 1371, 1383, 1527, 1563, 1623, 1671

Offset: 0

Antti Karttunen, Apr 04 2000

nonn

find_A055095_is_2_composites := proc(upto_n) local j,a; a := []; for j from 1 to upto_n do if(-1 = (j - wt(GrayCode(qrs2bincode((2*j)+1))))) then if(not isprime((2*j)+1)) then a := [op(a),((2*j)+1)]; fi; fi; od; RETURN(a); end;

A005811[n_] := Length[Length /@ Split[IntegerDigits[n, 2]]];
A055094[n_] := With[{rr = Table[Mod[k^2, n], {k, 1, n-1}] // Union}, Boole[MemberQ[rr, #]] & /@ Range[n-1]] // FromDigits[#, 2]&;
A055095[1] = 0; A055095[n_] := 2*A005811[A055094[n]] - (n-1);
A055131 = Position[Array[A055095, 2000], 2] // Flatten // Select[#, CompositeQ]& (* Jean-François Alcover, Mar 06 2016 *)

a(n) = 3*((4*A005098[n])+1) = 3*A002144[n] ??? (Conjecture, not yet proved)

More terms from James Sellers, Apr 21 2000

cp's OEIS Frontend

A089986 Numbers n such that 4n + 7 is prime.

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A173331 Second of two intermediate sequences for integral solution of A002144(n)=x^2+y^2.

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A096029 Values (x+y-1)/2, where x^2+y^2=p runs over the Pythagorean primes A002144.

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Crossrefs

Formula

Extensions

A096030 Values (y-x-1)/2, where x^2+y^2=p,(xA002144.

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Extensions

A111215 Numbers k such that 4k + 5 is prime.

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Examples

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Programs

Magma

Mathematica

PARI

Formula

A173330 First of two intermediate sequences for integral solution of A002144(n)=x^2+y^2.

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Author

Keywords

Comments

Examples

References

Crossrefs

Formula

A094896 If 4*n+1 is prime and 4*n+3 is not prime then a(n)=4*n+1, else a(n)=0.

Views

Author

Keywords

Crossrefs

Programs

Magma

Maple

Mathematica

A119681 Odd numbers n such that 2n-1 is prime.

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Author

Keywords

Comments

Programs

Mathematica

PARI

Formula

Extensions

A163366 a(n) = (-1)^floor((prime(n)+2)/2) mod prime(n).

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Author

A094896 If 4n+1 is prime and 4n+3 is not prime then a(n)=4*n+1, else a(n)=0.