A309433 - OEIS

%I #16 Sep 03 2024 12:24:49

%S 0,0,0,0,0,0,0,1,3,5,11,17,30,38,57,74,103,129,173,209,267,323,402,

%T 477,583,683,820,954,1125,1295,1515,1727,1995,2264,2590,2917,3316,

%U 3713,4188,4668,5229,5800,6470,7140,7918,8712,9618,10539,11590,12655,13862

%N Number of prime parts in the partitions of n into 6 parts.

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

%F a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_(i=j..floor((n-j-k-l-m)/2)} (A010051(i) + A010051(j) + A010051(k) + A010051(l) + A010051(m) + A010051(n-i-j-k-l-m)).

%t Table[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[n - i - j - k - l - m] - PrimePi[n - i - j - k - l - m - 1]), {i, j, Floor[(n - j - k - l - m)/2]}], {j, k, Floor[(n - k - l - m)/3]}], {k, l, Floor[(n - l - m)/4]}], {l, m, Floor[(n - m)/5]}], {m, Floor[n/6]}], {n, 0, 50}]

%t Table[Count[Flatten[IntegerPartitions[n,{6}]],_?PrimeQ],{n,0,50}] (* _Harvey P. Dale_, Sep 03 2024 *)

%Y Cf. A010051, A259196, A309427.

%K nonn

%O 0,9

%A _Wesley Ivan Hurt_, Aug 03 2019