A391762 Positive integers k such that both 2*k + 1 and 4*k^2 + 1 are composite, and either 2*k is a perfect power with odd exponent >= 3 or 2*k = 2^r with r > 2 and r not a power of 2.

4, 16, 32, 64, 108, 256, 500, 512, 864, 1024, 1372, 2048, 2916, 3888, 4000, 4096, 5324, 6912, 8192, 8788, 10976, 13500, 16384, 19652, 27436, 32000, 37044, 42592, 48668, 50000, 55296, 62500, 65536, 70304, 78732, 87808, 97556, 108000, 119164, 124416, 131072, 139968, 143748, 157216, 171500, 186624

Offset

1,1

Comments

All terms k in this sequence satisfy that (2*k)^m + 1 is composite for every m >= 1. Proof: (Start)

Let b = 2*k. Assume (i) 2*k + 1 and 4*k^2 + 1 are composite, and (ii) either b is a perfect power with odd exponent t >= 3, or b = 2^r with r > 2 and r not a power of 2 (so r has an odd factor s >= 3, hence b = (2^(r/s))^s).

Odd m: b^m + 1 is divisible by b + 1, which is composite by (i).

Even m: write m = 2^u*v with u >= 1 and v odd. If v >= 3 (so m is not a power of 2), then b^m + 1 = (b^(2^u))^v + 1 is divisible by b^(2^u) + 1. If v = 1 (so m = 2^u), then for u = 1 we have b^2 + 1 = 4*k^2 + 1 composite by (i); for u >= 2, by (ii) we can write b = c^t with odd t >= 3, hence b^(2^u) + 1 = (c^(2^u))^t + 1 is divisible by c^(2^u) + 1.

Therefore (2*k)^m + 1 is composite for all m >= 1. (End)

For k in A394571 that are not in this sequence: For some such k (e.g., 17, 22, 24, ...), a small j >= 2 making (2*k)^(2^j) + 1 prime is easy to find. For others (e.g., 19, 25, 31, ...), finding such a j may be difficult, or no such j may exist.

Links

Table of n, a(n) for n=1..46.

Example

4 is a term since 2*4 + 1 = 9 = 3^2 and 4*4^2 + 1 = 65 = 5*13 are both composite, and b = 2*4 = 8 = 2^3 is a power of 2 with r = 3 > 2, and 3 is not a power of 2.
108 is a term since 2*108 + 1 = 217 = 7*31 and 4*108^2 + 1 = 46657 = 13*37*97 are both composite, and b = 2*108 = 216 = 6^3 is a perfect power with odd exponent 3.

Maple

A391762 := proc(n)
    option remember;
    local k, p, r, b;
    if n = 1 then return 4; end if;
    for k from A391762(n - 1) + 1 do
        b := 2*k;
        if not isprime(b + 1) and not isprime(b^2 + 1) then
            if iperfpow(b, 'p') <> FAIL and type(p, integer) and p >= 3 and type(p, odd) then
                return k
            else
                r := ilog2(b);
                if 2^r = b and r > 2 and not is(r, 2^integer) then
                    return k
                end if
            end if
        end if
    end do;
end proc:
seq(A391762(n), n = 1 .. 56);

Crossrefs

Subsequence of A394571.

Cf. A000079, A000215, A001912, A002144, A002522, A008586, A047845, A070265, A094550, A121944.

Sequence in context: A383915 A326873 A126032 * A382325 A296819 A034713

Adjacent sequences: A391759 A391760 A391761 * A391763 A391765 A391766

Keyword

nonn

Author

Felix Huber, Apr 03 2026

Status

approved