A396958 a(n) = n - Omega(|Re((5 + 2*i)^n)|) - Omega(|Im((5 + 2*i)^n)|), where Omega is A001222.

-1, -3, -1, -3, 0, -4, 2, -3, 3, -1, 5, -2, 6, 3, 6, 0, 12, 2, 11, 4, 9, 9, 18, 3, 14, 12, 16, 13, 19, 9, 24, 10, 20, 24, 23, 12, 31, 23, 25, 16, 33, 19, 35, 24, 29, 34, 40, 17, 39, 28, 38, 31, 47, 27, 39, 30, 44, 42, 55, 29, 53, 46, 43, 38, 48, 38, 61, 50, 55

Offset

1,2

Comments

Let u(n) + i*v(n) = (5 + 2*i)^n, so that u(n)^2 + v(n)^2 = 29^n, where 29 == 1 (mod 4); such primes admit a representation as a sum of two squares. Then a(n) = n - Omega(u(n)) - Omega(v(n)) = Omega(u(n)^2 + v(n)^2) - Omega(u(n)) - Omega(v(n)). The values u(n) and v(n) can be computed via the recurrence u(n + 1) = 5*u(n) - 2*v(n), v(n + 1) = 2*u(n) + 5*v(n), with u(1) = 5 and v(1) = 2.

Since u(n)^2 + v(n)^2 = 29^n and gcd(u(n), v(n)) = 1 (any common divisor must divide 29^n), the triple (u(n)^2 - v(n)^2, 2*u(n)*v(n), u(n)^2 + v(n)^2) is a primitive Pythagorean triple for all n >= 1.

Thus a(n) compares the additive prime factor count of the norm 29^n with that of its Gaussian components u(n) and v(n).

Large positive values of a(n) occur when both u(n) and v(n) have relatively few prime factors.

Since |u(n)| and |v(n)| are of order about p^(n/2), probabilistic number theory suggests that Omega(u(n)) and Omega(v(n)) grow roughly like log log |u(n)|, which is about log(n). A refined heuristic suggests a(n) ~ n - 2*log(n) - C_p, where C_p = 2*(log(log(p)) - log(2) + B), with p = u(1)^2 + v(1)^2 and B the Meissel-Mertens constant (A077761). Fluctuations around this trend are of order sqrt(log(n)) and are governed by Erdős-Kac type behavior of Omega. For p = 29 this gives C_p ~ 1.56, explaining the stronger downward shift, see linked graph.

Links

Table of n, a(n) for n=1..69.

Felix Huber, Growth of a(n).

Eric Weisstein's World of Mathematics, Erdős-Kac theorem.

Eric Weisstein's World of Mathematics, Hardy-Ramanujan theorem.

Wikipedia, Fermat's theorem on sums of two squares.

Formula

a(n) = n - A001222(|Re((5 + 2*i)^n)|) - A001222(|Im((5 + 2*i)^n)|).

Example

a(2) = -3 since (5 + 2*i)^2 = 21 + 20*i, so Omega(21) = 2, Omega(20) = 3 and hence 2 - 2 - 3 = -3.
a(11) = 5 since (5 + 2*i)^11 = -55535695 - 95479298*i, so Omega(55535695) = 4, Omega(95479298) = 2 and hence 11 - 4 - 2 = 5.

Maple

with(NumberTheory):
A396958List := proc(N)
    local a, b, k, u, v;
    u := 5;
    v := 2;
    a := Vector(N);
    for k from 1 to N do
        a[k] := k - Omega(abs(u)) - Omega(abs(v));
        b := 5*u - 2*v;
        v := 2*u + 5*v;
        u := b;
    end do;
    convert(a, list)
end proc:
A396958List(69);

Mathematica

Table[n - PrimeOmega@ Abs@ Re[#] - PrimeOmega@ Abs@ Im[#] &[(5 + 2*I)^n], {n, 69}] (* Michael De Vlieger, Jun 18 2026 *)

Crossrefs

Cf. A000351, A001222, A002144, A077761, A396955, A396956, A396957, A396959.

Sequence in context: A349004 A127549 A191009 * A082648 A389526 A174233

Adjacent sequences: A396955 A396956 A396957 * A396959 A396960 A396962

Keyword

sign

Author

Felix Huber, Jun 17 2026

Status

approved