A023201 -id:A023201 - cp's OEIS frontend

A291455 Number of ways to write 2n+1 as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x + 3y + 5z + 7w, x^3 + 3y^3 + 5z^3 + 7w^3 and x^7 + 3y^7 + 5z^7 + 7w^7 are all prime.

3, 2, 5, 1, 1, 2, 5, 1, 5, 3, 3, 3, 3, 4, 6, 2, 5, 1, 3, 2, 6, 3, 2, 1, 4, 4, 6, 4, 2, 6, 2, 5, 8, 3, 1, 3, 4, 10, 7, 1, 2, 5, 5, 4, 5, 2, 2, 6, 7, 4, 2, 1, 4, 4, 4, 2, 6, 9, 8, 2, 4, 7, 12, 3, 4, 2, 1, 6, 7, 1, 4, 5, 8, 4, 10, 2, 5, 3, 7, 3, 8, 7, 3, 4, 6, 2, 5, 10, 6, 7, 3, 8, 10, 7, 3, 5, 4, 5, 7, 1, 6

Offset: 0

Zhi-Wei Sun, Aug 24 2017

nonn

Conjecture: (i) a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 3, 4, 7, 17, 23, 34, 39, 51, 66, 69, 99, 109, 115, 171, 191. Also, any integer n > 1 with gcd(n,42) = 1 can be written as x + 3*y + 5*z + 7*w with x,y,z,w nonnegative integers such that x^3 + 3*y^3 + 5*z^3 + 7*w^3 and x^7 + 3*y^7 + 5*z^7 + 7*w^7 are both prime.
(ii) Any positive odd integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x + 5*y + 9*z + 11*w, x^3 + 5*y^3 + 9*z^3 + 11*w^3 and x^5 + 5*y^5 + 9*z^5 + 11*w^5 are all prime. Also, any integer n > 1 with gcd(n,30) = 1 can be written as x + 5*y + 9*z + 11*w with x,y,z,w nonnegative integers such that x^3 + 5*y^3 + 9*z^3 + 11*w^3 and x^5 + 5*y^5 + 9*z^5 + 11*w^5 are both prime.
(iii) Let (k,m) be one of the ordered pairs (1,2), (1,4), (1,5), (1,9), (2,6), (3,5), (8,8). Then any positive odd integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x^k + 3*y^k + 5*z^k + 7*w^k and x^m + 3*y^m + 5*z^m + 7*w^m are both prime.
(iv) Any positive odd integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that p = x + 3*y + 5*z + 7*w and 2*p+1 (or p-4) are both prime.
(v) For each m = 1, 2, 4, any positive odd integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that p = x^m + 3*y^m + 5*z^m + 7*w^m and p+6 are both prime.
See also A290935 for a similar conjecture involving twin primes.

			a(4) = 1 since 2*4+1 = 0^2 + 2^2 + 2^2 + 1^2 with 0 + 3*2 + 5*2 + 7*1 = 23, 0^3 + 3*2^3 + 5*2^3 + 7*1^3 = 71 and 0^7 + 3*2^7 + 5*2^7 + 7*1^7 = 1031 all prime.
a(34) = 1 since 2*34+1 = 2^2 + 0^2 + 4^2 + 7^2 with 2 + 3*0 + 5*4 + 7*7 = 71, 2^3 + 3*0^3 + 5*4^3 + 7*7^3 = 2729 and 2^7 + 3*0^7 + 5*4^7 + 7*7^7 = 5846849 all prime.
a(66) = 1 since 2*66+1 = 4^2 + 6^2 + 9^2 + 0^2 with 4 + 3*6 + 5*9 + 7*0 = 67, 4^3 + 3*6^3 + 5*9^3 + 7*0^3 = 4357 and 4^7 + 3*6^7 + 5*9^7 + 7*0^7 = 24771037 all prime.
a(69) = 1 since 2*69+1 = 11^2 + 3^2 + 0^2 + 3^2 with 11 + 3*3 + 5*0 + 7*3 = 41, 11^3 + 3*3^3 + 5*0^3 + 7*3^3 = 1601 and 11^7 + 3*3^7 + 5*0^7 + 7*3^7 = 19509041 all prime.
a(191) = 1 since 2*191+1 = 11^2 + 6^2 + 1^2 + 15^2 with 11 + 3*6 + 5*1 + 7*15 = 139, 11^3 + 3*6^3 + 5*1^3 + 7*15^3 = 25609 and 11^7 + 3*6^7 + 5*1^7 + 7*15^7 = 1216342609 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.

Cf. A000040, A000118, A005384, A023201, A046132, A271518, A281976, A290935, A291150, A291191.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
f[m_,x_,y_,z_,w_]:=f[m,x,y,z,w]=x^m+3y^m+5z^m+7w^m;
Do[r=0;Do[If[SQ[2n+1-x^2-y^2-z^2]&&PrimeQ[f[1,x,y,z,Sqrt[2n+1-x^2-y^2-z^2]]]&&PrimeQ[f[3,x,y,z,Sqrt[2n+1-x^2-y^2-z^2]]]&&PrimeQ[f[7,x,y,z,Sqrt[2n+1-x^2-y^2-z^2]]],r=r+1],{x,0,Sqrt[2n+1]},{y,0,Sqrt[2n+1-x^2]},{z,0,Sqrt[2n+1-x^2-y^2]}];Print[n," ",r],{n,0,100}]

A307561 Numbers k such that both 6k - 1 and 6k + 5 are prime.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 9, 14, 17, 18, 22, 28, 29, 32, 38, 39, 42, 43, 44, 52, 58, 59, 64, 74, 77, 84, 93, 94, 98, 99, 107, 108, 109, 113, 137, 143, 147, 157, 158, 162, 163, 169, 182, 183, 184, 197, 198, 203, 204, 213, 214, 217, 227, 228, 238, 239, 247, 248, 249, 259, 267, 268, 269, 312, 317, 318, 329, 333, 344

Offset: 1

Sally Myers Moite, Apr 14 2019

nonn

There are 146 terms below 10^3, 831 terms below 10^4, 5345 terms below 10^5, 37788 terms below 10^6 and 280140 terms below 10^7.
Prime pairs differing by 6 are called "sexy" primes. Other prime pairs with difference 6 are of the form 6n + 1 and 6n + 7.
Numbers in this sequence are those which are not 6cd + c - d - 1, 6cd + c - d, 6cd - c + d - 1 or 6cd - c + d, that is, they are not (6c - 1)d + c - 1, (6c - 1)d + c, (6c + 1)d - c - 1 or (6c + 1)d - c.

			a(2) = 2, so 6(2) - 1 = 11 and 6(2) + 5 = 17 are both prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Sally M. Moite, "Primeless" Sieves for Primes and for Prime Pairs Which Differ by 2m, vixra:1903.0353 (2019).

Primes differing from each other by 6 are A023201, A046117.
Similar sequences for twin primes are A002822, A067611, for "cousin" primes A056956, A186243.
Intersection of A024898 and A059325.
Cf. also A307562, A307563.

Select[Range[500], PrimeQ[6# - 1] && PrimeQ[6# + 5] &] (* Alonso del Arte, Apr 14 2019 *)

is(k) = isprime(6*k-1) && isprime(6*k+5); \\ Jinyuan Wang, Apr 20 2019

A307562 Numbers k such that both 6k + 1 and 6k + 7 are prime.

Original entry on oeis.org

1, 2, 5, 6, 10, 11, 12, 16, 17, 25, 26, 32, 37, 45, 46, 51, 55, 61, 62, 72, 76, 90, 95, 100, 101, 102, 121, 122, 125, 137, 142, 146, 165, 172, 177, 181, 186, 187, 205, 215, 216, 220, 237, 241, 242, 247, 257, 270, 276, 277, 282, 290, 291, 292, 296, 297, 310, 311, 312, 331, 332, 335, 347, 355, 356, 380, 381, 390

Offset: 1

Sally Myers Moite, Apr 14 2019

nonn

There are 138 such numbers between 1 and 1000.
Prime pairs that differ by 6 are called "sexy" primes. Other prime pairs that differ by 6 are of the form 6n - 1 and 6n + 5.
Numbers in this sequence are those which are not 6cd - c - d - 1, 6cd - c - d, 6cd + c + d - 1 or 6cd + c + d, that is, they are not (6c - 1)d - c - 1, (6c - 1)d - c, (6c + 1)d + c - 1 or (6c + 1)d + c.

			a(3) = 5, so 6(5) + 1 = 31 and 6(5) + 7 = 37 are both prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Sally M. Moite, “Primeless” Sieves for Primes and for Prime Pairs Which Differ by 2m, vixra:1903.0353 (2019).

For the primes see A023201, A046117.
Similar sequences for twin primes are A002822, A067611, for "cousin" primes A056956, A186243.
Intersection of A024899 and A153218.
Cf. also A307561, A307563.

Select[Range[400], AllTrue[6 # + {1, 7}, PrimeQ] &] (* Michael De Vlieger, Apr 15 2019 *)

isok(n) = isprime(6*n+1) && isprime(6*n+7); \\ Michel Marcus, Apr 16 2019

A103576 Concatenations of pairs of primes that differ by 1000000.

Original entry on oeis.org

31000003, 371000037, 1511000151, 1931000193, 1991000199, 2111000211, 3131000313, 3671000367, 3971000397, 4091000409, 4571000457, 5411000541, 5471000547, 5771000577, 6191000619, 6911000691, 8291000829, 8591000859

Offset: 1

Jonathan Vos Post, Mar 23 2005

After the first element, 31000003, which is prime, integers in this sequence can never be prime, as they are all multiples of 3. They can be semiprimes, as is the case for 3671000367 = 3 x 1223666789, 4571000457 = 3 x 1523666819, 5411000541 = 3 x 1803666847, 9071000907 = 3 x 3023666969.

			Prime(47) = 211 and 211 + 1000000 = Prime(78515) = 1000211. Concatenating these two primes gives 2111000211 = 3^4 * 17^2 * 31 * 2909.

Cf. A000040, A001358, A023201, A100750, A103195, A103206, A104718, A104719, A103523, A103534.

a(n) = Concatenate(P, P+1000000) iff P prime and P+1000000 prime.

A103617 Concatenations of pairs of primes that differ by 10^9.

Original entry on oeis.org

71000000007, 971000000097, 1031000000103, 1811000000181, 2231000000223, 2411000000241, 2711000000271, 3491000000349, 4091000000409, 4331000000433, 4391000000439, 6071000000607, 6131000000613, 7871000000787, 8291000000829

Offset: 1

Jonathan Vos Post, Mar 25 2005

Integers in this sequence can never be prime, as they are all multiples of 3. They can be semiprimes, as is the case for Prime(42) concatenated with Prime(50847544) = 1811000000181 = 3 x 603666666727.

			181 is prime, 181+10^9 = 1000000181 is prime, so their concatenation is an element of this sequence: 1811000000181. Coincidentally, prime(181)+10^9 = 1000001087 is also prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A001358, A023201, A100750, A103195, A103206, A104718, A104719, A103523, A103534, A103576.

FromDigits[Join[IntegerDigits[#],IntegerDigits[#+10^9]]]&/@Select[Prime[ Range[ 200]],PrimeQ[ #+ 10^9]&] (* Harvey P. Dale, May 14 2022 *)

a(n) = Concatenate(P, P+1000000000) iff P prime and P+1000000000 prime.

A104010 Sum of two successive sexy primes.

Original entry on oeis.org

16, 20, 28, 32, 40, 52, 68, 80, 88, 100, 112, 128, 140, 152, 172, 200, 208, 212, 220, 268, 308, 320, 340, 352, 388, 392, 452, 460, 472, 508, 520, 532, 548, 560, 620, 628, 668, 700, 712, 740, 752, 772, 872, 892, 920, 928, 1012, 1088, 1120, 1132, 1148, 1180, 1192

Offset: 1

Giovanni Teofilatto, Mar 31 2005

Eric Weisstein's World of Mathematics, Sexy Primes. [The definition in this webpage is unsatisfactory, because it defines a "sexy prime" as a pair of primes.- _N. J. A. Sloane_, Mar 07 2021].

Cf. A023201, A046117.

A104010 := proc(n)
    2*A023201(n)+6 ;
end proc: # R. J. Mathar, May 28 2013

a(n)= A023201(n)+A046117(n) = 2*A087695(n). [From R. J. Mathar, Nov 26 2008]

20 added, 84 removed, extended by R. J. Mathar, Nov 26 2008

A104037 Numbers of primes between two sexy primes.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1

Offset: 1

Giovanni Teofilatto, Mar 31 2005

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Sexy Primes. [The definition in this webpage is unsatisfactory, because it defines a "sexy prime" as a pair of primes.- _N. J. A. Sloane_, Mar 07 2021].

Cf. A023201 (sexy primes), A046117.

p:=1: q:= 0: r:= 1: s:= 1: count:= 0: Res:= NULL:
while count < 100 do
  t:= charfcn[{true}](isprime(p+6));
  if t=1 and q=1 then
     count:= count + 1;
     Res:= Res, r+s;
  fi;
  p:= p+2;
  q:= r; r:= s; s:= t;
od:
Res; # Robert Israel, Jun 23 2019

Corrected and extended by Robert Israel, Jun 23 2019

A187758 Number of ways to write n=x+y (x,y>0) with 2x-3, 2x+3, 6y+1 and 6y+5 all prime.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 2, 2, 2, 3, 4, 2, 2, 3, 3, 3, 2, 3, 3, 4, 5, 3, 6, 5, 4, 6, 3, 5, 4, 3, 6, 2, 4, 5, 5, 4, 4, 6, 5, 4, 6, 5, 4, 5, 7, 5, 2, 3, 6, 4, 5, 4, 5, 7, 6, 9, 5, 4, 9, 5, 4, 5, 5, 4, 5, 6, 3, 8, 5, 8, 8, 3, 7, 5, 3, 5, 3, 5, 4, 9, 6, 4, 9, 7, 5, 8, 7, 8, 6, 9, 8, 2, 7, 7, 5, 6, 2, 10, 6, 3

Offset: 1

Zhi-Wei Sun, Jan 03 2013

nonn

Conjecture: a(n)>0 for all n>4.

This has been verified for n up to 10^8. It implies that there are infinitely many cousin primes and also infinitely many sexy primes.

			a(5)=1 since 5=4+1 with 2*4-3, 2*4+3, 6*1+1 and 6*1+5 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..20000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A023200, A046132, A023201, A002375, A187757, A199920, A219055, A218867, A220455.

a[n_]:=a[n]=Sum[If[PrimeQ[2k-3]==True&&PrimeQ[2k+3]==True&&PrimeQ[6(n-k)+1]==True&&PrimeQ[6(n-k)+5]==True,1,0],{k,1,n-1}]
Do[Print[n," ",a[n]],{n,1,100}]

A219966 Number of ways to write n=p+q+(n mod 2)q with q<=n/2 and p, q, q+6 all prime.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 3, 2, 2, 3, 2, 3, 3, 2, 1, 4, 3, 1, 4, 3, 1, 4, 2, 3, 3, 2, 2, 4, 3, 2, 4, 2, 2, 5, 3, 4, 5, 2, 1, 5, 3, 2, 4, 1, 1, 5, 4, 4, 4, 3, 2, 5, 3, 2, 4, 3, 4, 5, 3, 4, 6, 3, 3, 6, 3, 3, 8, 5, 2, 6, 3, 4, 6, 2, 2, 9, 5, 3, 5, 4, 2, 6, 4

Offset: 1

Zhi-Wei Sun, Dec 02 2012

nonn

Conjecture: a(n)>0 for all n>11.

This conjecture is stronger than Goldbach's conjecture and Lemoine's conjecture. It can be further strengthened; see A219055 and the comments there.

			a(19)=1 since 19=5+2*7 with 5, 7, 7+6 all prime.
a(20)=1 since 20=13+7 with 13, 7, 7+6 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A219055, A023201, A002375, A046927.

a[n_]:=a[n]=Sum[If[PrimeQ[Prime[k]+6]==True&&PrimeQ[n-(1+Mod[n,2])Prime[k]]==True,1,0],{k,1,PrimePi[n/2]}]
Do[Print[n," ",a[n]],{n,1,10000}]

A220951 Primes p such that p+6 is also prime and there is a power of two in the interval (p,p+6).

Original entry on oeis.org

5, 7, 11, 13, 31, 61, 251, 4093

Offset: 1

Brad Clardy, Feb 20 2013

A search for sexy primes bracketing a power of two was conducted up to 2^1500. It is conjectured that this is a finite sequence.

On the basis of existing work about primes of the form 2^n+k and 2^n-k, plus a few additional tests, we have a(9) > 2^750740. - Giovanni Resta, Feb 21 2013

Cf. A023201, A220746, A221211.

//Program finds primes separated by an even number (called gap) which have a power of two between them. Program starts with the smallest power of two above gap. Primes less than this starting point can be checked by inspection.
gap:=6;
start:=Ilog2(gap)+1;
for i:= start to 1000 do
   powerof2:=2^i;
   for k:=powerof2-gap+1 to powerof2-1 by 2 do
      if (IsPrime(k) and IsPrime(k+gap)) then
         k;
      end if;
   end for;
end for;

pptQ[n_]:=AllTrue[{n,n+6},PrimeQ]&&Count[Log[2,#]&/@Range[n,n+6], ?IntegerQ] > 0; Select[Range[4100],pptQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, Nov 01 2015 *)

cp's OEIS Frontend

A291455 Number of ways to write 2*n+1 as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x + 3*y + 5*z + 7*w, x^3 + 3*y^3 + 5*z^3 + 7*w^3 and x^7 + 3*y^7 + 5*z^7 + 7*w^7 are all prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A307561 Numbers k such that both 6*k - 1 and 6*k + 5 are prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A307562 Numbers k such that both 6*k + 1 and 6*k + 7 are prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A103576 Concatenations of pairs of primes that differ by 1000000.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A103617 Concatenations of pairs of primes that differ by 10^9.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A104010 Sum of two successive sexy primes.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

Extensions

A104037 Numbers of primes between two sexy primes.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Extensions

A187758 Number of ways to write n=x+y (x,y>0) with 2x-3, 2x+3, 6y+1 and 6y+5 all prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

A291455 Number of ways to write 2n+1 as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x + 3y + 5z + 7w, x^3 + 3y^3 + 5z^3 + 7w^3 and x^7 + 3y^7 + 5z^7 + 7w^7 are all prime.

A307561 Numbers k such that both 6k - 1 and 6k + 5 are prime.

A307562 Numbers k such that both 6k + 1 and 6k + 7 are prime.