A378887 a(n) = gcd(n, A001511(n)).

1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1

Offset

1,2

Links

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

Formula

a(n) = A378888(2*n) = gcd(2*n, A007814(2*n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} k * d(k) = 1.62390224555724623579..., where d(k) = Sum_{m>=0} phi(2*m+1)/((2*m+1)*odd(k)*2^(k*(2*m+1))) is the asymptotic density of the occurrences of k in this sequence, phi(k) = A000010(k) is the Euler totient function, and odd(k) = A000265(k) is the odd part of k. Equivalently, the asymptotic mean is Sum_{m>=0} (phi(2*m+1)/(2*m+1)) * Sum_{k>=1} 1/2^(k*(2*m+1)-v_2(k)), where v_2(k) = A007814(k).

Mathematica

a[n_] := GCD[n, IntegerExponent[n, 2] + 1]; Array[a, 100]

Prog

(PARI) a(n) = gcd(n, valuation(n, 2) + 1);

Crossrefs

Cf. A000010, A000265, A001511, A007814, A378888.

Sequence in context: A304750 A104306 A074389 * A353688 A353666 A051119

Adjacent sequences: A378884 A378885 A378886 * A378888 A378889 A378890

Keyword

nonn,easy

Author

Amiram Eldar, Dec 10 2024

Status

approved