A363327 Total number of parts coprime to n in the partitions of n into 9 parts.

0, 0, 0, 0, 0, 0, 0, 0, 9, 8, 18, 21, 45, 48, 88, 96, 198, 153, 369, 298, 542, 533, 1107, 673, 1695, 1375, 2280, 2017, 4392, 1962, 6588, 4668, 7337, 6695, 11924, 6703, 19746, 13220, 20416, 15497, 37854, 15056, 51372, 31987, 43358, 43930, 91404, 39701, 111571, 62525

Offset

1,9

Links

Table of n, a(n) for n=1..50.

Index entries for sequences related to partitions

Formula

a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} (c(i) + c(j) + c(k) + c(l) + c(m) + c(o) + c(p) + c(q) + c(n-i-j-k-l-m-o-p-q)), where c(x) = [gcd(n,x) = 1] and [ ] is the Iverson bracket.

Example

The partitions of 12 into 9 parts are: 1+1+1+1+1+1+1+1+4, 1+1+1+1+1+1+1+2+3, and 1+1+1+1+1+1+2+2+2. 12 is coprime to 1, but not 2, 3, or 4. Since there are 21 total parts in these partitions that are coprime to 12, a(12) = 21.

Mathematica

Table[Count[Flatten[IntegerPartitions[n, {9}]], _?(CoprimeQ[#, n]&)], {n, 50}] (* Harvey P. Dale, Sep 10 2024 *)

Crossrefs

For similar sequences into k parts for k = 2..10, see: A000010(n>2) (k=2), A363278 (k=3), A363322 (k=4), A363323 (k=5), A363324 (k=6), A363325 (k=7), A363326 (k=8), this sequence (k=9), A363328 (k=10).

Sequence in context: A231483 A309656 A370882 * A309657 A296615 A340446

Adjacent sequences: A363324 A363325 A363326 * A363328 A363329 A363330

Keyword

nonn,easy

Author

Wesley Ivan Hurt, May 27 2023

Status

approved