A168534 -id:A168534 - cp's OEIS frontend

A078392 Sum of GCD's of parts in all partitions of n.

1, 3, 5, 9, 11, 20, 21, 35, 42, 61, 66, 112, 113, 168, 210, 279, 313, 461, 508, 719, 852, 1088, 1277, 1756, 2006, 2573, 3106, 3937, 4593, 5958, 6872, 8676, 10305, 12655, 15009, 18664, 21673, 26559, 31447, 38217, 44623, 54386, 63303, 76379, 89696, 106879

Offset: 1

Vladeta Jovovic, Dec 24 2002

nonn

Equals row sums of triangle A168534. - Gary W. Adamson, Nov 28 2009

			Partitions of 4 are 1+1+1+1, 1+1+2, 2+2, 1+3, 4, the corresponding GCD's of parts are 1,1,2,1,4 and their sum is a(4) = 9.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000010, A000041, A168534, A181844 (the same for LCM), A319301.

with(numtheory): with(combinat):
a:= n-> add(phi(n/d)*numbpart(d), d=divisors(n)):
seq(a(n), n=1..50);  # Alois P. Heinz, Apr 02 2015

a[n_] := Sum[EulerPhi[n/d]*PartitionsP[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Jul 01 2015, after Alois P. Heinz *)

a(n) = Sum_{d|n} d * A000837(n/d).
a(n) = Sum_{d|n} phi(n/d)*numbpart(d) = Sum_{d|n} A000010(n/d)*A000041(d). - Vladeta Jovovic, May 08 2003
From Richard L. Ollerton, May 06 2021: (Start)
a(n) = Sum_{k=1..n} A000041(gcd(n,k)).
a(n) = Sum_{k=1..n} A000041(n/gcd(n,k))*A000010(gcd(n,k))/A000010(n/gcd(n,k)). (End)

A168533 Triangle read by rows, A000012 * A168532, as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 1, 4, 1, 1, 7, 2, 1, 1, 13, 2, 1, 1, 1, 20, 4, 2, 1, 1, 1, 34, 4, 2, 1, 1, 1, 1, 51, 7, 2, 2, 1, 1, 1, 1, 78, 7, 4, 2, 1, 1, 1, 1, 1, 112, 13, 4, 2, 2, 2, 21, 1, 1, 167, 13, 4, 2, 2, 1, 1, 1, 1, 1, 1, 230, 20, 7, 4, 2, 2, 21, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Nov 28 2009

Row sums = A026905: (1, 3, 6, 11, 18, 29, 44, 66,...)

Triangle A168534 = A168532 * A000012

			First few rows of the triangle =
1;
2, 1;
4, 1, 1;
7, 2, 1, 1;
13, 2, 1, 1, 1;
20, 4, 2, 1, 1, 1;
34, 4, 2, 1, 1, 1, 1;
51, 7, 2, 2, 1, 1, 1, 1;
78, 7, 4, 2, 1, 1, 1, 1, 1;
112, 13, 4, 2, 2, 1, 1, 1, 1, 1;
167, 13, 4, 2, 2, 1, 1, 1, 1, 1, 1;
230, 20, 7, 4, 2, 2, 1, 1, 1, 1, 1, 1;
330, 20, 7, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1;
449, 34, 7, 4, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1;
616, 34, 13, 4, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1;
825, 51, 13, 7, 4, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1;
1121, 51, 13, 7, 4, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1;

Cf. A168532, A026905, A168534

Triangle read by rows, A000012 * A168532; where A000012 = an infinite lower
triangular matrix with all 1's. The operation takes partial sums of A168532
column terms.

cp's OEIS Frontend

A078392 Sum of GCD's of parts in all partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula