A389957 Even perfect powers plus 1.

5, 9, 17, 33, 37, 65, 101, 129, 145, 197, 217, 257, 325, 401, 485, 513, 577, 677, 785, 901, 1001, 1025, 1157, 1297, 1445, 1601, 1729, 1765, 1937, 2049, 2117, 2305, 2501, 2705, 2745, 2917, 3137, 3365, 3601, 3845, 4097, 4357, 4625, 4901, 5185, 5477, 5777, 5833, 6085

Offset

1,1

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = A075090(n) + 1.

a(n) = A389765(n) + 2.

a(n) = A389956(n) / A389764(n).

Sum_{n>=1} 1/a(n) = Pi^2/12 + log(2) - 1 = A072691 + A002162 - 1.

Mathematica

seq[lim_] := Union[Table[m^k + 1, {k, 2, Log2[lim]}, {m, 2, Surd[lim, k], 2}] // Flatten]; seq[6000]

Prog

(PARI) list(lim) = {my(s = List()); for(k = 2, logint(lim, 2), forstep(m = 2, sqrtnint(lim, k), 2, listput(s, m^k + 1))); Set(s); }
(Python)
from sympy import mobius, integer_nthroot
from oeis_sequences.OEISsequences import bisection
def A389957(n): return bisection(lambda x:int(n+x+sum(mobius(k)*(integer_nthroot(x-1, k)[0]>>1) for k in range(2, (x-1).bit_length()))), n, n) # Chai Wah Wu, Oct 21 2025

Crossrefs

Intersection of A005408 and A216765.

Complement of A389958 within A216765.

Cf. A002162, A072691, A075090, A389764, A389765, A389956.

Sequence in context: A233187 A160426 A301786 * A258411 A059743 A000322

Adjacent sequences: A389954 A389955 A389956 * A389958 A389959 A389960

Keyword

nonn,easy

Author

Amiram Eldar, Oct 20 2025

Status

approved