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.

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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

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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.
		

Crossrefs

A359447 a(n) is the least number that is the sum of two cubes of primes and is 2^n times an odd prime, or -1 if there is no such number.

Original entry on oeis.org

-1, -1, 152, 2224, 9056, 108736, -1, 4532992, 34674176, 268684288, 2280249344, 18693763072, 138890141696, 1111848828928, 8803419521024, 70375767212032, 564861779443712, 4507018424221696, 36030079546425344, 288238419152207872, 2305850719072157696, 18446757709572210688, 147573952867129622528
Offset: 1

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Author

Robert Israel, Jan 01 2023

Keywords

Comments

a(n) is the least term of A086119 such that a(n)/2^n is an odd prime, or -1 if there is no such term.
Since p^3 + q^3 = (p+q)*(p^2 - p*q + q^2), we must have p+q = 2^n, and p^2 - p*q + q^2 an odd prime.
Is a(n) > 0 for all n > 7?

Examples

			a(3) = 152 because 3^3 + 5^3 = 152 = 2^3 * 19, 3 and 5 are primes and 19 is odd, and no smaller number works.
		

Crossrefs

Programs

  • Maple
    f:= proc(n) local p,q,t;
      t:= 2^n; p:= nextprime(t/2);
      while p > 2 do
        p:= prevprime(p);
        q:= t - p;
        if isprime(q) and isprime(p^2 - p*q + q^2) then return p^3 + q^3 fi
      od;
    -1
    end proc:
    map(f, [$1..20]);
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