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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. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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.

LINKS

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.

EXAMPLE

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.

MATHEMATICA

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}]

CROSSREFS

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

Sequence in context: A083254 A068453 A111986 * A248849 A172216 A121490

Adjacent sequences:  A291452 A291453 A291454 * A291456 A291457 A291458

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Aug 24 2017

STATUS

approved

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Last modified July 9 16:50 EDT 2020. Contains 335545 sequences. (Running on oeis4.)