A359448 a(n) is the least number that is the sum of two cubes of primes and is 2^n times an odd number.

35, 54, 468, 152, 16, 9056, 81088, 527744, 4532992, 33900032, 268684288, 2148866048, 17185288192, 137439174656, 1099611160576, 8797884612608, 70369850097664, 562950041894912, 4503607335190528, 36028810622664704, 288230406982991872, 2305843633483415552, 18446744212436156416, 147573952867129622528

Offset: 0

Author

Robert Israel, Jan 01 2023

Keywords

• nonn

Comments

a(n) is the least member k of A086119 such that A007814(k) = n.

a(n) <= A359447(n) if A359447(n) > 0.

Since p^3 + q^3 = (p+q)*(p^2 - p*q + q^2), except for n=4 we must have A007814(p+q) = n.

There is no analogous sequence for squares, because if p and q are odd primes p^2 + q^2 == 2 (mod 4).

Examples

			a(0) = 35 = 2^3 + 3^3 = 2^0 * 35 with 2 and 3 prime and 35 odd.
a(1) = 54 = 3^3 + 3^3 = 2^1 * 27 with 3 and 3 prime and 27 odd.
a(2) = 468 = 5^3 + 7^3 = 2^2 * 117 with 5 and 7 prime and 117 odd.
a(3) = 152 = 3^3 + 5^3 = 2^3 * 19 with 3 and 5 prime and 19 odd.
a(4) = 16 = 2^3 + 2^3 = 2^4 * 1 with 2 and 2 prime and 1 odd.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A007814, A086119, A359447.

Programs

Maple

f:= proc(n) local p,q,b,t,r;
  r:= infinity;
  for b from 1 by 2 while 2^(3*n-2)*b^3 < r do
    t:= 2^n*b;
    p:= nextprime(t/2);
    while p > 3 do
      p:= prevprime(p);
      q:= t-p;
      if p^3 + q^3 > r then break fi;
      if isprime(q) then r:= p^3 + q^3; break fi;
    od
  od;
    r
end proc:
f(0):= 35: f(4):= 16:
map(f, [$0..30]);