A363327 - OEIS

%I #9 Sep 10 2024 12:58:18

%S 0,0,0,0,0,0,0,0,9,8,18,21,45,48,88,96,198,153,369,298,542,533,1107,

%T 673,1695,1375,2280,2017,4392,1962,6588,4668,7337,6695,11924,6703,

%U 19746,13220,20416,15497,37854,15056,51372,31987,43358,43930,91404,39701,111571,62525

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

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F 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.

%e 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.

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

%Y 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).

%K nonn,easy

%O 1,9

%A _Wesley Ivan Hurt_, May 27 2023