A387569 a(n) is the smallest positive integer y such that there exists an integer x such that binomial(x + i*y, n) is a Gaussian integer.

1, 2, 3, 12, 6, 6, 42, 168, 252, 84, 924, 2772, 924, 924, 2772, 3696, 924, 5544, 52668, 105336, 52668, 105336, 1211364

Offset

1,2

Comments

Whether binomial(a+b*i, n) is a Gaussian integer appears to be periodic for fixed integer b, n (which means that it suffices to check up to a certain integer a before it begins cycling). It appears that all terms in the sequence are products of only ramified primes (2) and inert primes (3, 7, and other primes that are equivalent to 3 modulo 4). It would be great to find a generalized, explicit formula for any n.

Links

Table of n, a(n) for n=1..23.

Eric Weisstein's World of Mathematics, Gaussian Integer.

Rick Zhou, Even and Odd Gaussian Integer Lines

Example

For n = 1, b = 1 suffices, as binomial(0 + i, 1) = i is a Gaussian integer.
For n = 2, binomial(0 + 2*i, 2) = -2-i is a Gaussian integer.
For n = 3, the minimal b is 3, as binomial(1 + 3*i, 3) = -5*i is a Gaussian integer and no binomial coefficients with smaller b work.

Prog

(PARI) isok(n, b) = for (a=0, b, my(z=binomial(a+I*b, n)); if ((denominator(real(z))==1) && (denominator(imag(z))==1), return(1)));
a(n) = my(b=1); while (!isok(n, b), b++); b; \\ Michel Marcus, Dec 30 2025

Crossrefs

Sequence in context: A056819 A124442 A220271 * A088611 A361323 A292265

Adjacent sequences: A387566 A387567 A387568 * A387570 A387571 A387572

Keyword

nonn,more

Author

Rick Zhou, Dec 28 2025

Extensions

a(9)-a(18) from Michel Marcus, Dec 29 2025

a(19) from Michel Marcus, Dec 30 2025

a(20)-a(23) from Michel Marcus, Dec 31 2025

Status

approved