VOOZH about

URL: https://oeis.org/A355199

⇱ A355199 - OEIS

login
A355199
Sum of all the parts in the partitions of n into exactly 3 prime parts.
3
0, 0, 0, 0, 0, 0, 6, 7, 8, 18, 10, 22, 24, 26, 14, 45, 32, 68, 36, 57, 40, 105, 44, 115, 72, 125, 78, 189, 84, 203, 60, 186, 96, 297, 68, 280, 144, 333, 152, 390, 80, 451, 126, 430, 176, 540, 138, 611, 192, 588, 250, 765, 208, 848, 162, 770, 280, 969, 174, 944, 240, 976, 372
OFFSET
0,7
FORMULA
a(n) = n * Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} c(i) * c(j) * c(n-i-j), where c = A010051.
a(n) = n * A068307(n) for n > 0.
a(n) = A355196(n) + A355197(n) + A355198(n).
EXAMPLE
a(9) = 18; since 9 can be written as the sum of 3 primes in two ways: 2+2+5 = 3+3+3 and the sum of all the parts of these partitions is (2+2+5)+(3+3+3) = 18, we have a(9) = 18.
MATHEMATICA
Table[n*Sum[Sum[(PrimePi[i] - PrimePi[i - 1]) (PrimePi[j] - PrimePi[j - 1]) (PrimePi[n - i - j] - PrimePi[n - i - j - 1]), {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 23 2022
STATUS
approved