A319301 -id:A319301 - 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)

A306956 Sum over all partitions of n into distinct parts of the LCM of the parts.

Original entry on oeis.org

1, 1, 2, 5, 7, 15, 21, 39, 58, 90, 142, 218, 325, 465, 695, 948, 1411, 1977, 2883, 3940, 5415, 7422, 10126, 14091, 18947, 25666, 34282, 45890, 60710, 82211, 108510, 142960, 185271, 240595, 315158, 409231, 531967, 688689, 880997, 1126451, 1447754, 1849743

Offset: 0

Alois P. Heinz, Mar 17 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200
Wikipedia, Partition (number theory)

Cf. A000009, A000041, A001560, A040051, A052002, A181844, A319301.

b:= proc(n, i, r) option remember; `if`(i*(i+1)/2 b(n$2, 1):
seq(a(n), n=0..44);

b[n_, i_, r_] := b[n, i, r] = If[i(i+1)/2 < n, 0, If[n == 0, r, b[n, i-1, r] + b[n-i, Min[i-1, n-i], LCM[i, r]]]];
a[n_] := b[n, n, 1];
Table[a[n], {n, 0, 44}] (* Jean-François Alcover, Mar 20 2019, translated from Maple *)

a(n) mod 2 = A040051(n).
a(n) is even <=> n in { A001560 }.
a(n) is odd  <=> n in { A052002 }.

A319300 Irregular triangle where T(n,k) is the number of strict integer partitions of n with GCD equal to the k-th divisor of n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 2, 1, 0, 1, 4, 1, 4, 1, 0, 1, 6, 1, 1, 7, 2, 0, 1, 11, 1, 10, 2, 1, 1, 0, 1, 17, 1, 17, 4, 0, 1, 23, 2, 1, 1, 26, 4, 1, 0, 1, 37, 1, 36, 6, 2, 1, 0, 1, 53, 1, 53, 7, 2, 1, 0, 1, 70, 4, 1, 1, 77, 11, 0, 1, 103, 1, 103, 10, 4, 2, 1

Offset: 1

Gus Wiseman, Sep 16 2018

			Triangle begins:
   1
   0  1
   1  1
   1  0  1
   2  1
   2  1  0  1
   4  1
   4  1  0  1
   6  1  1
   7  2  0  1
  11  1
  10  2  1  1  0  1
  17  1
  17  4  0  1
  23  2  1  1
  26  4  1  0  1
  37  1
  36  6  2  1  0  1
  53  1
  53  7  2  1  0  1

A regular version is A303138. Row lengths are A000005. Row sums are A000009. First column is A078374.

Cf. A000837, A027750, A289508, A289509, A303140, A303280, A319299, A319301.

Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,GCD@@#==k]&]],{n,20},{k,Divisors[n]}]

T(n,k) = A078374(n/A027750(n,k)).

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

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A306956 Sum over all partitions of n into distinct parts of the LCM of the parts.

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A319300 Irregular triangle where T(n,k) is the number of strict integer partitions of n with GCD equal to the k-th divisor of n.

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