A309439 Number of prime parts in the partitions of n into 10 parts.

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, 698, 924, 1176, 1521, 1909, 2418, 2991, 3729, 4560, 5610, 6795, 8254, 9906, 11919, 14180, 16908, 19972, 23615, 27706, 32527, 37917, 44227, 51267, 59425, 68525, 79007

Offset

0,13

Links

Table of n, a(n) for n=0..50.

Index entries for sequences related to partitions

Formula

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)).

Mathematica

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

Crossrefs

Cf. A010051, A259201.

Sequence in context: A309436 A309437 A309438 * A108542 A006450 A085918

Adjacent sequences: A309436 A309437 A309438 * A309440 A309441 A309442

Keyword

nonn

Author

Wesley Ivan Hurt, Aug 03 2019

Status

approved