A389956 Odd numbers that are one less than the square of a perfect power.

15, 63, 255, 1023, 1295, 4095, 9999, 16383, 20735, 38415, 46655, 65535, 104975, 159999, 234255, 262143, 331775, 456975, 614655, 809999, 999999, 1048575, 1336335, 1679615, 2085135, 2559999, 2985983, 3111695, 3748095, 4194303, 4477455, 5308415, 6249999, 7311615, 7529535, 8503055, 9834495

Offset

1,1

Links

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

Formula

a(n) = A389765(n) * A389957(n).

Sum_{n>=1} 1/a(n) = 1/2 - Pi^2/24 = 1/2 - A222171 = A354238 / 2.

Mathematica

seq[lim_] := Union[Table[m^(2*k) - 1, {k, 2, Log2[lim + 1]/2}, {m, 2, Surd[lim + 1, 2*k], 2}] // Flatten]; seq[10^7]

Prog

(PARI) list(lim) = {my(s = List()); for(k = 2, logint(lim+1, 2)\2, forstep(m = 2, sqrtnint(lim+1, 2*k), 2, listput(s, m^(2*k) - 1))); Set(s); }
(Python)
from math import isqrt
from sympy import mobius, integer_nthroot
from oeis_sequences.OEISsequences import bisection
def A389956(n): return bisection(lambda x:int(n+x+sum(mobius(k)*(integer_nthroot(isqrt(x+1), k)[0]>>1) for k in range(2, isqrt(x+1).bit_length()))), n, n) # Chai Wah Wu, Oct 21 2025

Crossrefs

Intersection of A005408 and A062965.

Complement of A389955 within A062965.

Cf. A001597, A222171, A354238, A389765, A389957.

Sequence in context: A305616 A104473 A135972 * A138104 A152099 A119496

Adjacent sequences: A389953 A389954 A389955 * A389957 A389958 A389959

Keyword

nonn,easy

Author

Amiram Eldar, Oct 20 2025

Status

approved