%I #14 Oct 04 2019 14:17:34

%S 0,0,0,0,0,1,3,5,11,12,19,23,32,36,47,53,68,77,92,103,123,134,157,173,

%T 197,216,245,265,299,323,357,385,425,454,499,534,580,619,671,711,770,

%U 816,875,926,993,1044,1116,1175,1249,1314,1396,1462,1552,1625,1714

%N Number of prime parts in the partitions of n into 4 parts.

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

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

%e Figure 1: The partitions of n into 4 parts for n = 8, 9, ..

%e 1+1+1+9

%e 1+1+2+8

%e 1+1+3+7

%e 1+1+4+6

%e 1+1+1+8 1+1+5+5

%e 1+1+2+7 1+2+2+7

%e 1+1+1+7 1+1+3+6 1+2+3+6

%e 1+1+2+6 1+1+4+5 1+2+4+5

%e 1+1+3+5 1+2+2+6 1+3+3+5

%e 1+1+1+6 1+1+4+4 1+2+3+5 1+3+4+4

%e 1+1+1+5 1+1+2+5 1+2+2+5 1+2+4+4 2+2+2+6

%e 1+1+2+4 1+1+3+4 1+2+3+4 1+3+3+4 2+2+3+5

%e 1+1+3+3 1+2+2+4 1+3+3+3 2+2+2+5 2+2+4+4

%e 1+2+2+3 1+2+3+3 2+2+2+4 2+2+3+4 2+3+3+4

%e 2+2+2+2 2+2+2+3 2+2+3+3 2+3+3+3 3+3+3+3

%e --------------------------------------------------------------------------

%e n | 8 9 10 11 12 ...

%e --------------------------------------------------------------------------

%e a(n) | 11 12 19 23 32 ...

%e --------------------------------------------------------------------------

%e - _Wesley Ivan Hurt_, Sep 08 2019

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

%t Table[Count[Flatten[IntegerPartitions[n,{4}]],_?PrimeQ],{n,0,60}] (* _Harvey P. Dale_, Oct 04 2019 *)

%Y Cf. A010051, A026810.

%K nonn

%O 0,7

%A _Wesley Ivan Hurt_, Aug 01 2019