A390066 Integers which can be represented by a^b + c^d where a,b,c,d are distinct primes.

174, 246, 253, 292, 294, 364, 368, 371, 375, 412, 414, 464, 486, 512, 532, 604, 632, 654, 704, 772, 872, 966, 1084, 1086, 1184, 1204, 1304, 1356, 1363, 1380, 1459, 1494, 1500, 1612, 1620, 1692, 1712, 1806, 1860, 1924, 1974, 2024, 2092, 2172, 2173, 2192, 2212, 2219

Offset

1,1

Comments

No odd integer can be represented in more than one way.

The smallest term not to use the prime 2 is a(277) = 18138 = A390179(1).

Links

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Example

292 = 3^5 + 7^2
4938 = 17^3 + 5^2
18138 = 7^5 + 11^3

Prog

(PARI) is_a390066(t) = {forprime(p1=2, sqrtint(t), my(lim2=logint(t, p1)); forprime(p2=2, lim2, if(p2==p1, next); my(p=p1^p2, r=t-p); if(r<8, next); my(f=factor(r)); if(omega(f)==1 && f[1, 1]!=p1 && f[1, 1]!=p2, my(p4=bigomega(f)); if(isprime(p4) && p4!=p1 && p4!=p2 && p4!=f[1, 1], return(1))))); 0} \\ Hugo Pfoertner, Oct 28 2025
(Python)
from sympy import nextprime, integer_nthroot
aupto = 10000000
A390066, parts, p1 = set(), [], 2
while p1 < integer_nthroot(aupto, 2)[0]:
    p2 = 2
    while (pot:=p1**p2) < aupto:
        if p1 != p2: parts.append([pot, p1, p2])
        p2 = nextprime(p2)
    p1 = nextprime(p1)
parts = sorted(parts)
for pf in parts:
    for ps in parts:
        if (p:=pf[0] + ps[0]) > aupto: break
        else:
            if pf[1] not in ps[1:] and pf[2] not in ps[1:]: A390066.add(p)
print((A390066:=(sorted(A390066)))[0:50]) # Karl-Heinz Hofmann, Nov 11 2025

Crossrefs

Subsequence of A298591.

Cf. A390178, A390179, A390188.

Sequence in context: A377875 A045257 A025381 * A168349 A179136 A249432

Adjacent sequences: A390063 A390064 A390065 * A390067 A390068 A390069

Keyword

nonn,easy

Author

Paul Duckett, Oct 23 2025

Status

approved