A363278 Total number of parts coprime to n in the partitions of n into 3 parts.

0, 0, 3, 2, 6, 3, 12, 8, 15, 10, 30, 12, 42, 21, 32, 32, 72, 27, 90, 40, 66, 55, 132, 48, 130, 78, 126, 84, 210, 60, 240, 128, 170, 136, 216, 108, 342, 171, 240, 160, 420, 126, 462, 220, 276, 253, 552, 192, 525, 250, 416, 312, 702, 243, 560, 336, 522, 406, 870, 240, 930, 465

Offset

1,3

Links

Table of n, a(n) for n=1..62.

Index entries for sequences related to partitions

Formula

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} ([gcd(n,n-i-j) = 1] + [gcd(n,i) = 1] + [gcd(n,j) = 1]), where [ ] is the Iverson Bracket.

Example

The partitions of 5 into 3 parts are: 3+1+1 and 2+2+1. 5 is coprime to 1, 2 and 3. Since there are 6 total parts in the partitions of 5 that are coprime to 5, a(5) = 6.

Mathematica

Table[Sum[Sum[KroneckerDelta[GCD[n, n - i - j], 1] + KroneckerDelta[GCD[n, j], 1] + KroneckerDelta[GCD[n, i], 1], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 100}]

Crossrefs

For similar sequences into k parts for k = 2..10, see: A000010(n>2) (k=2), this sequence (k=3), A363322 (k=4), A363323 (k=5), A363324 (k=6), A363325 (k=7), A363326 (k=8), A363327 (k=9), A363328 (k=10).

Sequence in context: A064455 A141619 A270143 * A065021 A048652 A195345

Adjacent sequences: A363275 A363276 A363277 * A363279 A363280 A363281

Keyword

nonn,easy

Author

Wesley Ivan Hurt, May 25 2023

Status

approved