A168534 -id:A168534

Search: a168534 -id:a168534

Displaying 1-2 of 2 results found.

page 1

A078392

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

+10
13

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

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

FORMULA

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)

EXAMPLE

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.

MAPLE

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

MATHEMATICA

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 *)

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Vladeta Jovovic, Dec 24 2002

STATUS

approved

A168533

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

+10
1

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

Triangle A168534 = A168532 * A000012

LINKS

Table of n, a(n) for n=1..77.

FORMULA

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.

EXAMPLE

First few rows of the triangle =

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;

CROSSREFS

Cf. A168532, A026905, A168534

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Nov 28 2009

STATUS

approved

page 1

Search completed in 0.010 seconds