A045637 -id:A045637 - cp's OEIS frontend

A078597 Primes of the form p*(p+4)+2 where p and p+4 are primes.

23, 79, 223, 439, 4759, 53359, 77839, 95479, 99223, 159199, 194479, 239119, 378223, 416023, 680623, 2223079, 2595319, 2873023, 3186223, 3515623, 4003999, 5022079, 6456679, 6859159, 8732023, 9235519, 9492559, 10017223, 10595023

Offset: 1

Cino Hilliard, Dec 08 2002

More generally, if a and b are even numbers, let Seq(a,b) be the sequence of primes of the form p*(p+a)+b where p and p+a are primes. Seq(a,b) is finite if either a^2+b == 2 (mod 3) or a^2-4*b is a square. Is it infinite in all other cases?

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Except for the term 23, this is a subsequence of A048880. A051779 is Seq(2, 2). A049002 is Seq(0, -2). A045637 is Seq(0, 4).

Select[ #(#+4)+2&/@Select[Prime/@Range[500], PrimeQ[ #+4]&], PrimeQ]

prodtp(n1,n2,a,b)=local(f,x); f=0; forprime(x=n1,n2,if(isprime(x+a),f=x*(x+a)+b; if(isprime(f),print(x" "x+a" "f" "); ); ); ); \ Computes that part of Seq(a,b) with n1<=p<=n2.

Edited by Dean Hickerson, Dec 10 2002

A107312 Primes p such that p + 2 and p^2 + 2^2 are primes.

Original entry on oeis.org

3, 5, 17, 137, 347, 827, 2087, 2687, 3557, 3917, 4517, 4967, 5477, 5657, 5867, 6827, 7457, 7547, 7877, 8087, 8537, 8597, 10037, 10427, 10937, 12107, 12377, 13397, 13877, 16067, 17837, 17987, 19427, 19697, 20507, 20717, 20807, 22367, 22637

Offset: 1

Zak Seidov, May 21 2005

nonn

Primes are lesser twins. Except a(1) and a(2), all a(n) == 7(mod 10).

Edward Jiang, Table of n, a(n) for n = 1..1000

Cf. A045637.

[p: p in PrimesUpTo(25000)|  IsPrime(p+2) and IsPrime(p^2+4)]; // Vincenzo Librandi, Jan 29 2011

select(p -> isprime(p) and isprime(p+2) and isprime(p^2+4), [seq(2*i+1,i=1..10000)]); # Robert Israel, Aug 11 2014

Select[Prime[Range[3000]], PrimeQ[ #+2]&&PrimeQ[ #^2+4]&]

a(n)=isprime(n) && isprime(n+2) && isprime(n^2+4) \\ Edward Jiang, Aug 08 2014

A174050 Primes of the form x^2 + y^2 such that L(x)* L(y) = 1, where L is the Liouville lambda-function A008836.

Original entry on oeis.org

2, 13, 17, 29, 37, 53, 73, 89, 97, 101, 113, 173, 181, 193, 197, 233, 241, 257, 277, 293, 313, 337, 349, 353, 373, 409, 421, 433, 449, 457, 521, 541, 569, 577, 593, 613, 641, 661, 673, 677, 709, 733, 757, 761, 809, 821, 853, 881, 929, 1021, 1033, 1049, 1069

Offset: 1

Michel Lagneau, Mar 06 2010

nonn

One contribution to the set of solutions is from (x,y) where x and y are both prime, see A045637.

Another set of solutions is contributed if (x,y) are both in A026424.

			2 is in the sequence because 2 = 1 + 1 and L(1)*L(1)= (1) *(1) = 1.
13 is in the sequence because 13 = 2^2 + 3^2 and L(2)*L(3)= (-1)*(-1) = 1.
193 is in the sequence because 193 = 12^2 + 7^2 and L(12)*L(7)= (-1)*(-1) = 1.

Cf. A002819, A002313, A174054.

isA174050 := proc(n)
        local x,y ;
        if not isprime(n) then
                return false;
        end if;
        for x from 1 do
                if x^2 > n then
                        return false;
                end if;
                if issqr(n-x^2) then
                        y := sqrt(n-x^2) ;
                        if A008836(x) * A008836(y) = 1 then
                                return true;
                        end if;
                end if;
        end do:
end proc:
for n from 1 to 1100 do
        if isA174050(n) then
                printf("%d,\n",n) ;
        end if;
end do: # R. J. Mathar, Jul 09 2012

lambdaQ[{x_, y_}] := LiouvilleLambda[x]*LiouvilleLambda[y] == 1; Select[ Prime /@ Range[200], Or @@ lambdaQ /@ PowersRepresentations[#, 2, 2] &] (* Jean-François Alcover, Jul 30 2013 *)

A182476 Primes of the form p^2+100, where p is prime.

Original entry on oeis.org

109, 149, 269, 389, 461, 941, 1061, 1949, 2309, 2909, 3581, 3821, 10301, 10709, 11549, 11981, 16229, 18869, 19421, 22901, 24749, 26669, 30029, 32141, 44621, 52541, 57221, 72461, 76829, 94349, 96821, 109661, 128981, 134789, 167381, 201701, 214469, 253109

Offset: 1

Alex Ratushnyak, May 01 2012

Cf. A045637 (p^2 + 4 is prime), A079141 (p^2 + 6 is prime), A182475.

Select[Table[p^2 + 100, {p, Prime[Range[200]]}], PrimeQ] (* T. D. Noe, May 01 2012 *)

A217717 Primes of the form x^2 + y^2 - 1, where x and y are primes.

Original entry on oeis.org

7, 17, 73, 97, 193, 241, 313, 337, 409, 457, 577, 1009, 1129, 1201, 1249, 1321, 1489, 1657, 1801, 1873, 2017, 2137, 2377, 2521, 2689, 2833, 2857, 3049, 3169, 3217, 3361, 3529, 3697, 3769, 3889, 4057, 4177, 4441, 4513, 4561, 4657, 5209, 5449, 5569, 5689, 5857

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, Mar 21 2013

nonn

Unlike primes of the form x^2+y^2 (A045637) which can be redefined as x^2+4, and primes of the form x^2+y^2+1 (A182475) which can be redefined as primes of the form x^2+10, this sequence appears to have no one-variable analog. In the preceding, x and y are prime.

			457 is in the sequence because it is a prime number, and 457 = 13^2 + 17^2 - 1.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Cf. A045637 (primes of the form p^2+4, where p is prime).

Cf. A182475 (primes of the form p^2+10, where p is prime).

mx = 25; Union[Select[Flatten[Table[Prime[a]^2 + Prime[b]^2 - 1, {a, mx}, {b, a, mx}]], # < Prime[mx]^2 && PrimeQ[#] &]] (* T. D. Noe, Mar 29 2013 *)

A287924 Numbers k such that A287922(k) is a prime.

Original entry on oeis.org

2, 3, 4, 6, 7, 12, 15, 19, 21, 25, 27, 33, 38, 39, 44, 51, 59, 62, 63, 65, 66, 69, 74, 90, 93, 96, 101, 106, 108, 111, 112, 123, 132, 138, 143, 144, 147, 153, 162, 163, 166, 168, 181, 187, 188, 203, 219, 224, 229, 241, 243, 255, 258, 259, 269, 273, 300

Offset: 1

XU Pingya, Jun 02 2017

nonn

Corresponding primes are in A045637.

Cf. A287922, A045637.

Select[Table[n, {n, 300}], PrimeQ[Prime[#]^2 + 4] &]

A339692 Primes that can be expressed as p^k+2*k where p is prime and k >= 1.

Original entry on oeis.org

5, 7, 13, 19, 29, 31, 43, 53, 61, 73, 89, 103, 109, 131, 139, 151, 173, 181, 193, 199, 229, 241, 271, 283, 293, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1373, 1429, 1453, 1483, 1489, 1609

Offset: 1

J. M. Bergot and Robert Israel, Dec 13 2020

nonn

Terms expressible in more than one way include
  13 = 11^1 + 2*1 = 3^2 + 2*2
  349 = 347^1 + 2*1 = 7^3 + 2*3
  78139 = 78137^^1 + 2*1 = 5^7 + 2*7
  1092733 = 1092731^1 + 2*1 = 103^3 + 2*3
  22665193 = 22665191^1 + 2*1 = 283^3 + 2*3.

			a(5) = 29 is a term because 29 = 5^2 + 2*2. and 5 and 29 are primes.

Robert Israel, Table of n, a(n) for n = 1..10000

Includes A006512, A045637 and A201308.

N:= 1000: # for terms <= N
S:= {}:
for n from 1 while 3^n + 2*n <= N do
  p:= 2:
  do
    p:= nextprime(p);
    q:=  p^n + 2*n;
    if q > N then break fi;
    if isprime(q) then S:= S union {q};
    fi
od od:
sort(convert(S,list));

Block[{nn = 1610, a = {}}, Do[Do[Which[# > nn, Break[], PrimeQ[#], AppendTo[a, #]] &[(#^k) + 2 k], {k, Infinity}] &[Prime@ i], {i, 2, PrimePi@ nn}]; Union@ a] (* Michael De Vlieger, Dec 13 2020 *)

isok(p) = {if (isprime(p), for(k=1, p\2, if (k==isprimepower(p-2*k), return(1));););} \\ Michel Marcus, Dec 13 2020

A129119 Numbers of the form 2*p (with p a prime number) such that p^2+4 is prime.

Original entry on oeis.org

6, 10, 14, 26, 34, 74, 94, 134, 146, 194, 206, 274, 326, 334, 386, 466, 554, 586, 614, 626, 634, 694, 746, 926, 974, 1006, 1094, 1154, 1186, 1214, 1226, 1354, 1486, 1574, 1646, 1654, 1706, 1766, 1906, 1934, 1966, 1994, 2174, 2234, 2246, 2474, 2734, 2846

Offset: 1

Giovanni Teofilatto, May 25 2007

			7^2 + 4 = 53 which is a prime number. Therefore 2*7 is in the sequence.

Cf. A045637, A062324.

2*Select[Prime@Range[250], PrimeQ[ #^2 + 4] &] (* Ray Chandler, May 27 2007 *)
a={};For[n=1,n<300,n++,If[PrimeQ[Prime[n]^2 + 4], AppendTo[a, 2*Prime[n]]]]; a (* Stefan Steinerberger, May 27 2007 *)

a(n) = 2*A062324(n).

Extended and edited by Ray Chandler and Stefan Steinerberger, May 27 2007

A162152 Numbers of the form x(x-1) + y(y-1) with x^2 + y^2 being a prime, x,y >= 0.

Original entry on oeis.org

0, 2, 8, 12, 22, 30, 32, 44, 50, 62, 76, 84, 90, 96, 98, 122, 132, 140, 158, 162, 174, 182, 212, 222, 240, 246, 254, 260, 274, 288, 292, 312, 326, 328, 348, 362, 372, 380, 386, 392, 404, 422, 432, 482, 490, 510, 524, 536, 552, 562, 572, 578, 582, 612, 618, 630, 638, 650

Offset: 1

Juri-Stepan Gerasimov, Jun 26 2009, Jul 29 2009

nonn

Each term is the sum of two oblong numbers and therefore even.

			a(1)=0 from x=y=1, with 2 a prime.
a(2)=2 from x=1, y=2, with 5 a prime.
a(3)=8 from x=2, y=3, with 13 a prime.
a(4)=12 from x=1, y=4, with 17 a prime.
a(4)=22 from x=2, y=5, with 29 a prime.
a(.)=212 from x=2, y=15, with 229 a prime, or from x=8, y=13, with 233 a prime.

Cf. A000290, A002313, A005843, A045637.

obl := proc(n) n*(n-1) ; end: lim := 800; L := {} ;
for x from 0 to lim/2 do for y from x to lim/2 do if obl(x)+obl(y) <= lim then if isprime(x^2+y^2) then L := L union { obl(x)+obl(y) } ; fi; fi; od: od: sort(L) ; # R. J. Mathar, Sep 11 2009

Take[#[[1]](#[[1]]-1)+#[[2]](#[[2]]-1)&/@Select[Tuples[ Range[ 0,40],2],PrimeQ[ Total[#^2]]&]//Union,60] (* Harvey P. Dale, Jun 07 2020 *)

Duplicates of 212 and 432 removed, 500 removed by R. J. Mathar, Sep 11 2009

A177831 Values of q in A176983.

Original entry on oeis.org

5, 29, 53, 173, 293, 1373, 2213, 4493, 5333, 9413, 10613, 18773, 26573, 27893, 37253, 54293, 76733, 78977, 85853, 94253, 97973, 100493, 120413, 139133, 151337, 177257, 192737, 212557, 214373, 237173, 249017, 253013, 299213, 326057, 332933, 351653

Offset: 1

Zak Seidov, May 14 2010

nonn

Includes all terms of A045637 beyond the first, since unless 3 | p either p^2 + 2 or p^2 + 4 must be divisible by 3.

Cf. A045637, A176983.

a(n) = nextprime(A176983(n)^2).

cp's OEIS Frontend

A078597 Primes of the form p*(p+4)+2 where p and p+4 are primes.

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A107312 Primes p such that p + 2 and p^2 + 2^2 are primes.

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A174050 Primes of the form x^2 + y^2 such that L(x)* L(y) = 1, where L is the Liouville lambda-function A008836.

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A182476 Primes of the form p^2+100, where p is prime.

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A217717 Primes of the form x^2 + y^2 - 1, where x and y are primes.

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A287924 Numbers k such that A287922(k) is a prime.

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A339692 Primes that can be expressed as p^k+2*k where p is prime and k >= 1.

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A129119 Numbers of the form 2*p (with p a prime number) such that p^2+4 is prime.

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