A078392 -id:A078392 - cp's OEIS frontend

A181846 Triangle read by rows: T(n,k) = Sum_{c in P(n,n-k+1)} gcd(c) where P(n,m) = A008284(n,m) is the number of partitions of n into m parts.

1, 1, 2, 1, 1, 3, 1, 1, 3, 4, 1, 1, 2, 2, 5, 1, 1, 2, 4, 6, 6, 1, 1, 2, 3, 4, 3, 7, 1, 1, 2, 3, 6, 6, 8, 8, 1, 1, 2, 3, 5, 6, 9, 6, 9, 1, 1, 2, 3, 5, 8, 10, 10, 11, 10, 1, 1, 2, 3, 5, 7, 10, 11, 10, 5, 11, 1, 1, 2, 3, 5, 7, 12, 14, 19, 19, 17, 12, 1, 1, 2, 3, 5, 7, 11, 14, 18, 18, 14, 6, 13

Offset: 1

Peter Luschny, Dec 07 2010

See A181842 for the definition of 'partition'.

			[1]   1
[2]   1   2
[3]   1   1   3
[4]   1   1   3   4
[5]   1   1   2   2   5
[6]   1   1   2   4   6   6
[7]   1   1   2   3   4   3   7

Cf. A078392.

with(combstruct):
a181846_row := proc(n) local k,L,l,R,part;
R := NULL;
for k from 1 to n do
   L := 0;
   part := iterstructs(Partition(n),size=n-k+1):
   while not finished(part) do
      l := nextstruct(part);
      L := L + igcd(op(l));
   od;
   R := R,L;
od;
R end:

T[n_, k_] := GCD @@@ IntegerPartitions[n, {n-k+1}] // Total;
Table[T[n, k], {n, 1, 13}, {k, 1, n}] (* Jean-François Alcover, Jun 22 2019 *)

Extended to 13 rows by Jean-François Alcover, Jun 22 2019

A333697 a(n) = Sum_{d|n} (-1)^omega(n/d) * phi(rad(n/d)) * p(d), where p = A000041 (partition numbers).

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 9, 14, 22, 31, 46, 59, 89, 114, 158, 201, 281, 337, 472, 570, 756, 936, 1233, 1456, 1926, 2323, 2942, 3556, 4537, 5334, 6812, 8088, 10021, 11997, 14805, 17432, 21601, 25507, 30971, 36606, 44543, 52106, 63219, 74097, 88680, 104281, 124708, 145205, 173429, 202124

Offset: 1

Ilya Gutkovskiy, Apr 02 2020

nonn

Cf. A000010, A000041, A001221, A007947, A008683, A023900, A047968, A078392.

Table[Sum[(-1)^PrimeNu[n/d] EulerPhi[Last[Select[Divisors[n/d], SquareFreeQ]]] PartitionsP[d], {d, Divisors[n]}], {n, 50}]

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
a(n) = sumdiv(n, d, (-1)^omega(n/d) * eulerphi(rad(n/d)) * numbpart(d)); \\ Michel Marcus, Apr 03 2020

a(n) = Sum_{d|n} A023900(n/d) * A000041(d).
a(n) = Sum_{d|n} A047968(n/d) * mu(d) * d.
Sum_{k=1..n} a(gcd(n,k)) = A000041(n).

cp's OEIS Frontend

A181846 Triangle read by rows: T(n,k) = Sum_{c in P(n,n-k+1)} gcd(c) where P(n,m) = A008284(n,m) is the number of partitions of n into m parts.

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