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A078392
Sum of GCD's of parts in all partitions of n.
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
OFFSET
1,2
COMMENTS
Equals row sums of triangle A168534. - Gary W. Adamson, Nov 28 2009
LINKS
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.
Sequence in context: A282098 A034760 A070639 * A364808 A187753 A339638
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Dec 24 2002
STATUS
approved