A361930 a(n) is the greatest prime p such that p + q^2 + r^3 = prime(n)^4 for some primes q and r.

29, 613, 2389, 14629, 28549, 83227, 130171, 279707, 707131, 923509, 1873867, 2825749, 3418627, 4879669, 7890347, 12116969, 13845473, 20150467, 25411387, 28398229, 38949947, 47458027, 62740901, 88529269, 104060267, 112550869, 131079589, 141157867, 163046969, 260144491, 294499771, 352275349

Offset

2,1

Comments

Conjecture: a(n) exists for all n > 2.

Links

Robert Israel, Table of n, a(n) for n = 2..10000

Example

a(2) = 29 because 29 + 5^2 + 3^3 = 81 = prime(2)^4, with 29, 5 and 3 prime.
a(3) = 613 because 613 + 2^2 + 2^3 = 625 = prime(3)^4. with 613, 2 and 2 prime.

Maple

Qs:= select(isprime, [$2..floor(sqrt(100000))]):
Rs:= select(isprime, [$2..floor(100000^(1/3))]):
QRs:= sort(select(t -> t[1]^2 + t[2]^3 < 100000, [seq(seq([q, r], q=Qs), r=Rs)]),
 (a, b) -> a[1]^2 + a[2]^3 < b[1]^2 + b[2]^3):
f:= proc(t) local t4, qr, p;
   t4:= t^4;
   for qr in QRs do
      p:= t4 - qr[1]^2 - qr[2]^3;
      if isprime(p) then return p fi
   od;
   FAIL
end proc:
seq(f(ithprime(i)), i=2..50);

Crossrefs

Cf. A030514 (prime(n)^4).

Sequence in context: A287050 A142521 A032630 * A014929 A006136 A199038

Adjacent sequences: A361927 A361928 A361929 * A361931 A361932 A361933

Keyword

nonn

Author

Zak Seidov and Robert Israel, Mar 30 2023

Status

approved