cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 21-25 of 25 results.

A283359 Numbers of the form p^2 + q^2 = r^3 + s^3 with p, q, r, s primes.

Original entry on oeis.org

370, 7202, 36650, 1345682, 2127890, 2685962, 2715410, 3872090, 4331090, 4657490, 6379130, 7887458, 12235970, 14386538, 17938730, 19909370, 22588130, 22665530, 22694978, 30027170, 30080258, 31576970, 39707642, 40024010, 42567698, 42735530, 48438290, 54517538, 62572970, 72096050
Offset: 1

Views

Author

Zak Seidov, Mar 06 2017

Keywords

Comments

Starting with a(2)=7202 all terms are congruent to 2 mod 24.
Intersection of A045636 and A086119.

Examples

			370=3^2+19^2=3^3+7^3, 7202=59^2+61^2=7^3+19^3.

Links

Crossrefs

Cf. A045636, A086119.

A359439 a(n) is the least number of the form p^2 + q^2 - 2 for primes p and q that is an odd multiple of 2^n, or -1 if there is no such number.

Original entry on oeis.org

11, 6, -1, 56, 16, 32, 192, 128, 2816, 1536, 15360, 30720, 12288, 73728, 147456, 32768, 196608, 1179648, 22806528, 11010048, 34603008, 31457280, 314572800, 679477248, 50331648, 301989888, 1006632960, 10871635968, 20132659200, 4831838208, 28991029248, 173946175488, 450971566080, 77309411328
Offset: 0

Views

Author

Robert Israel, Jan 02 2023

Keywords

Comments

Suggested by an email from J. M. Bergot.
a(2) = -1 because if p and q are odd primes, p^2 + q^2 - 2 is divisible by 8.

Examples

			a(0) = 11 = 2^2 + 3^2 - 2 = 11*2^0.
a(1) = 6 = 2^2 + 2^2 - 2 = 3*2^1.
a(3) = 56 = 3^2 + 7^2 - 2 = 7*2^3.
a(4) = 16 = 3^2 + 3^2 - 2 = 1*2^4.

Crossrefs

Cf. A045636

Programs

  • Maple
    f:= proc(n) local b,t,s,x,y;
    t:= 2^n;
    for b from 1 by 2  do
      if ormap(s -> subs(s,x) <= subs(s,y) and isprime(subs(s,x)) and isprime(subs(s,y)), [isolve(x^2+y^2-2=b*t)]) then return b*t fi
    od;
end proc:
f(2):= -1:
map(f, [$0..40]);

A286836 Even numbers that are the sum of two odd prime cubes.

Original entry on oeis.org

54, 152, 250, 370, 468, 686, 1358, 1456, 1674, 2224, 2322, 2540, 2662, 3528, 4394, 4940, 5038, 5256, 6244, 6886, 6984, 7110, 7202, 8190, 9056, 9826, 11772, 12194, 12292, 12510, 13498, 13718, 14364, 17080, 19026, 24334, 24416, 24514, 24732, 25720, 26586, 29302
Offset: 1

Views

Author

XU Pingya, Jul 31 2017

Keywords

Comments

Subsequence of A003325.

Crossrefs

Cf. A030078, A000578, A003325, A024670, A045636, A120398.

Programs

  • Mathematica
    Do[If[Prime[i]^3 + Prime[j]^3 == 2n, Print[2n]], {n, 15000}, {i, 2, n^(1/3)}, {j, i, (2n - i^3)^(1/3)}]

A357439 Sums of squares of two odd primes.

Original entry on oeis.org

18, 34, 50, 58, 74, 98, 130, 146, 170, 178, 194, 218, 242, 290, 298, 314, 338, 370, 386, 410, 458, 482, 530, 538, 554, 578, 650, 698, 722, 818, 850, 866, 890, 962, 970, 986, 1010, 1058, 1082, 1130, 1202, 1250, 1322, 1370, 1378, 1394, 1418, 1490, 1538, 1658, 1682
Offset: 1

Views

Author

Giuseppe Melfi, Oct 06 2022

Keywords

Comments

Although this is twice A143850, it is important enough to warrant an entry of it own. - N. J. A. Sloane, Oct 10 2022

Crossrefs

Cf. A045636, A103739, A143850, A227697.

A359492 a(n) is the least number of the form p^2 + q^2 - 2 for primes p and q that is an odd prime times 2^n, or -1 if there is no such number.

Original entry on oeis.org

11, 6, -1, 56, 48, 96, 192, 384, 2816, 1536, 109568, 10582016, 12288, 7429922816, 64176128, 4318724096, 196608, 60486975488, 9388028592128, 849566088298496, 214058289594368, 896029329195008
Offset: 0

Views

Author

Robert Israel, Jan 02 2023

Keywords

Comments

If a(n) > -1 then a(n) >= A359439(n).
a(22) <= 10228945815339008; a(23) <= 188039754665689088; a(24) <= 54409680373415936; a(25) <= 246561971023904768; a(26) <= 966464636658384896. - Daniel Suteu, Jan 05 2023

Examples

			a(4) = 48 = 3*2^4 = 5^2 + 5^2 - 2.

Crossrefs

Cf. A045636, A359439.

Programs

  • Maple
    f:= proc(n) local b,t,s,x,y;
    t:= 2^n; b:= 2;
    do
      b:= nextprime(b);
      if member(3, numtheory:-factorset(b*t+2) mod 4) then next fi;
      if ormap(s -> isprime(subs(s,x)) and isprime(subs(s,y)), [isolve(x^2+y^2-2=b*t)]) then return b*t fi
    od;
end proc:
f(2):= -1:
map(f, [$0..18]);

Extensions

a(19)-a(21) from Daniel Suteu, Jan 05 2023
Previous Showing 21-25 of 25 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.