A309439 - OEIS

%I #7 Dec 26 2020 18:16:01

%S 0,0,0,0,0,0,0,0,0,0,0,1,3,5,11,17,30,45,72,104,157,210,298,396,537,

%T 698,924,1176,1521,1909,2418,2991,3729,4560,5610,6795,8254,9906,11919,

%U 14180,16908,19972,23615,27706,32527,37917,44227,51267,59425,68525,79007

%N Number of prime parts 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)} (A010051(r) + A010051(q) + A010051(p) + A010051(o) + A010051(m) + A010051(l) + A010051(k) + A010051(j) + A010051(i) + A010051(n-i-j-k-l-m-o-p-q-r)).

%t Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[(PrimePi[i] - PrimePi[i - 1]) + (PrimePi[j] - PrimePi[j - 1]) + (PrimePi[k] - PrimePi[k - 1]) + (PrimePi[l] - PrimePi[l - 1]) + (PrimePi[m] - PrimePi[m - 1]) + (PrimePi[o] - PrimePi[o - 1]) + (PrimePi[p] - PrimePi[p - 1]) + (PrimePi[q] - PrimePi[q - 1]) + (PrimePi[r] - PrimePi[r - 1]) + (PrimePi[n - i - j - k - l - m - o - p - q - r] - PrimePi[n - i - j - k - l - m - o - p - q - r - 1]), {i, j, Floor[(n - j - k - l - m - o - p - q - r)/2]}], {j, k, Floor[(n - k - l - m - o - p - q - r)/3]}], {k, l, Floor[(n - l - m - o - p - q - r)/4]}], {l, m, Floor[(n - m - o - p - q - r)/5]}], {m, o, Floor[(n - o - p - q - r)/6]}], {o, p, Floor[(n - p - q - r)/7]}], {p, q, Floor[(n - q - r)/8]}], {q, r, Floor[(n - r)/9]}], {r, Floor[n/10]}], {n, 0, 50}]

%t Table[Count[Flatten[IntegerPartitions[n,{10}]],_?PrimeQ],{n,0,50}] (* _Harvey P. Dale_, Dec 26 2020 *)

%Y Cf. A010051, A259201.

%K nonn

%O 0,13

%A _Wesley Ivan Hurt_, Aug 03 2019