A364722 Numbers k that divide 1 + 2^m + 4^m for some m.

1, 3, 7, 13, 19, 21, 37, 39, 49, 57, 61, 67, 73, 79, 91, 97, 103, 109, 111, 139, 147, 151, 163, 169, 181, 183, 193, 199, 201, 211, 219, 237, 241, 271, 273, 291, 307, 309, 313, 327, 331, 337, 343, 349, 361, 367, 373, 379, 409, 417, 421, 427, 433, 453, 463, 469, 487, 489, 507, 523, 541, 543, 547

Offset

1,2

Comments

If k = 3*j+1 is prime and 2^j - 1 is not divisible by k, then k is a term, as 1 + 2^j + 4^j = (2^(k-1)-1)/(2^j - 1) == 0 (mod k). - Robert Israel, Aug 06 2023

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

a(4) = 13 is a term because 1 + 2^4 + 4^4 = 273 = 21 * 13 is divisible by 13.

Maple

filter:= proc(n) local x, r;
  for r in map(t -> subs(t, x), [msolve(1+x+x^2, n)]) do
    try
      NumberTheory:-ModularLog(r, 2, n);
    catch "no solutions exist": next
    end try;
    return true
  od;
  false
end proc:
select(filter, [seq(i, i=1..1000, 2)]);

Prog

(Python)
from itertools import count, islice
from sympy import sqrt_mod_iter, discrete_log
def A364722_gen(startvalue=1): # generator of terms >= startvalue
    if startvalue <= 1:
        yield 1
    if startvalue <= 3:
        yield 3
    for k in count(max(startvalue, 4)):
        for d in (r>>1 for r in sqrt_mod_iter(-3, k) if r&1):
            try:
                discrete_log(k, d, 2)
            except:
                continue
            yield k
            break
A364722_list = list(islice(A364722_gen(), 20)) # Chai Wah Wu, May 02 2024

Crossrefs

Subset of A034017. Cf. A364724.

Sequence in context: A034017 A034021 A216516 * A310260 A310261 A038978

Adjacent sequences: A364719 A364720 A364721 * A364723 A364724 A364725

Keyword

nonn

Author

Robert Israel, Aug 04 2023

Status

approved