|
%I #7 May 27 2023 23:21:53
%S 0,0,0,0,0,0,0,0,0,10,10,17,30,42,64,86,150,142,300,277,471,502,970,
%T 669,1556,1345,2190,2037,4230,2142,6530,4876,7657,7162,12746,7488,
%U 21120,14751,22864,17986,42420,18156,58880,38177,52533,53185,109360,49563,137515,79738
%N Total number of parts coprime to n in the partitions of n into 10 parts.
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = Sum_{r=1..floor(n/10)} Sum_{q=r..floor((n-r)/9)} Sum_{p=q..floor((n-q-r)/8)} Sum_{o=p..floor((n-p-q-r)/7)} Sum_{m=o..floor((n-o-p-q-r)/6)} Sum_{l=m..floor((n-m-o-p-q-r)/5)} Sum_{k=l..floor((n-l-m-o-p-q-r)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q-r)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q-r)/2)} (c(i) + c(j) + c(k) + c(l) + c(m) + c(o) + c(p) + c(q) + c(r) + c(n-i-j-k-l-m-o-p-q-r)), where c(x) = [gcd(n,x) = 1] and [ ] is the Iverson bracket.
%e The partitions of 13 into 10 parts are: 1+1+1+1+1+1+1+1+1+4, 1+1+1+1+1+1+1+1+2+3, and 1+1+1+1+1+1+1+2+2+2. 13 is coprime to 1, 2, 3, and 4. Since there are 30 total parts in these partitions that are coprime to 13, a(13) = 30.
%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), A363327 (k=9), this sequence (k=10).
%K nonn,easy
%O 1,10
%A _Wesley Ivan Hurt_, May 27 2023
|
