A359448 - OEIS

%I #15 Jul 09 2024 19:12:25

%S 35,54,468,152,16,9056,81088,527744,4532992,33900032,268684288,

%T 2148866048,17185288192,137439174656,1099611160576,8797884612608,

%U 70369850097664,562950041894912,4503607335190528,36028810622664704,288230406982991872,2305843633483415552,18446744212436156416,147573952867129622528

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

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

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

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

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

%H Robert Israel, <a href="/A359448/b359448.txt">Table of n, a(n) for n = 0..1000</a>

%e a(0) = 35 = 2^3 + 3^3 = 2^0 * 35 with 2 and 3 prime and 35 odd.

%e a(1) = 54 = 3^3 + 3^3 = 2^1 * 27 with 3 and 3 prime and 27 odd.

%e a(2) = 468 = 5^3 + 7^3 = 2^2 * 117 with 5 and 7 prime and 117 odd.

%e a(3) = 152 = 3^3 + 5^3 = 2^3 * 19 with 3 and 5 prime and 19 odd.

%e a(4) = 16 = 2^3 + 2^3 = 2^4 * 1 with 2 and 2 prime and 1 odd.

%p f:= proc(n) local p,q,b,t,r;

%p   r:= infinity;

%p   for b from 1 by 2 while 2^(3*n-2)*b^3 < r do

%p     t:= 2^n*b;

%p     p:= nextprime(t/2);

%p     while p > 3 do

%p       p:= prevprime(p);

%p       q:= t-p;

%p       if p^3 + q^3 > r then break fi;

%p       if isprime(q) then r:= p^3 + q^3; break fi;

%p     od

%p   od;

%p     r

%p end proc:

%p f(0):= 35: f(4):= 16:

%p map(f, [$0..30]);

%Y Cf. A007814, A086119, A359447.

%K nonn

%O 0,1

%A _Robert Israel_, Jan 01 2023