A378053 a(n) = gcd(Product_{d|n} (d + 1), Product_{d|n, d>1} (d - 1)) = gcd(A020696(n), A377484(n)).

1, 1, 2, 3, 4, 2, 2, 3, 16, 36, 2, 30, 4, 6, 16, 45, 4, 80, 2, 108, 16, 6, 2, 210, 24, 12, 32, 18, 4, 1008, 2, 45, 64, 12, 48, 8400, 4, 18, 16, 2268, 4, 240, 2, 90, 512, 18, 2, 3150, 32, 216, 64, 540, 4, 160, 144, 2430, 32, 12, 2, 166320, 4, 6, 1280, 405, 48, 1344

Offset: 1

Author

Amiram Eldar, Nov 15 2024

Keywords

• nonn

Links

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

Crossrefs

Cf. A000079, A002144, A002145, A020696, A377484, A378056.

Programs

Mathematica

a[n_] := GCD[Times @@ ((d = Divisors[n]) + 1), Times @@ (Rest@ d - 1)]; Array[a, 70]

PARI

a(n) = if(n == 1, 1, my(d = divisors(n)); gcd(prod(k=1, #d, d[k]+1), prod(k=2, #d, d[k]-1)));

Formula

a(n) = 2 if and only if n = 6 or n is a prime of the form 4*k+3 (A002145).

a(n) = 4 if and only if n is a prime of the form 4*k+1 (A002144).

a(n) == 1 (mod 2) if and only if n is a power of 2 (A000079).