A234307 a(n) = Sum_{i=1..n} gcd(2*n-i, i).

1, 3, 6, 8, 11, 17, 16, 20, 27, 31, 26, 44, 31, 45, 60, 48, 41, 75, 46, 80, 87, 73, 56, 108, 85, 87, 108, 116, 71, 165, 76, 112, 141, 115, 158, 192, 91, 129, 168, 196, 101, 239, 106, 188, 261, 157, 116, 256, 175, 235, 222, 224, 131, 297, 256, 284, 249, 199

Offset

1,2

Comments

Sum of the GCD's of the smallest and largest parts in the partitions of 2n into exactly two parts.

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Index entries for sequences related to partitions.

Formula

a(n) = Sum_{i=1..n} gcd(2*n-i, i).

a(n) = (A018804(2*n)-n)/2. - Sebastian Karlsson, Oct 03 2021

Conjecture: a(n) = (1/4)*Sum_{k = 1..4*n} (-1)^k *gcd(k, 8*n). - Peter Bala, Jan 01 2024

Sum_{k=1..n} a(k) ~ (Pi^2/4)*n^2 * (log(n) + 2*gamma - 1/2 + log(2)/6 - Pi^2/16 - zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). - Amiram Eldar, Mar 30 2024

Example

a(6) = 17; the partitions of 2(6) = 12 into two parts are: (11,1),(10,2),(9,3),(8,4),(7,5),(6,6). Then a(6) = gcd(11,1) + gcd(10,2) + gcd(9,3) + gcd(8,4) + gcd(7,5) + gcd(6,6). = 1 + 2 + 3 + 4 + 1 + 6 = 17.

Maple

A234307:=n->add( gcd(2*n-i, i), i=1..n); seq(A234307(n), n=1..100);

Mathematica

Table[Sum[GCD[2n - i, i], {i, n}], {n, 100}]
f[p_, e_] := (e*(p - 1)/p + 1)*p^e; a[n_] := (Times @@ f @@@ FactorInteger[2*n] - n)/2; Array[a, 100] (* Amiram Eldar, Apr 28 2023 *)

Prog

(PARI) a(n) = sum(i=1, n, gcd(i, 2*n-i)); \\ Michel Marcus, Dec 23 2013
(PARI) a(n) = {my(f = factor(2*n)); (prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; p^(e-1)*(p+e*(p-1))) - n)/2; } \\ Amiram Eldar, Mar 30 2024

Crossrefs

Cf. A001105 (sum of parts), A002378 (differences of parts).

Cf. A001620, A018804, A306016.

Sequence in context: A188018 A360532 A289241 * A175769 A352211 A160277

Adjacent sequences: A234304 A234305 A234306 * A234308 A234309 A234310

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Dec 22 2013

Status

approved