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A355198
Sum of the smallest parts of the partitions of n into exactly 3 prime parts.
3
0, 0, 0, 0, 0, 0, 2, 2, 2, 5, 2, 5, 4, 6, 2, 10, 4, 13, 4, 11, 4, 20, 4, 18, 6, 21, 6, 28, 6, 31, 4, 30, 6, 46, 4, 41, 8, 49, 8, 58, 4, 64, 6, 64, 8, 73, 6, 80, 8, 80, 10, 98, 8, 114, 6, 106, 10, 124, 6, 124, 8, 124, 12, 136, 6, 172, 10, 164, 12, 170, 4, 193, 10, 183, 12, 189
OFFSET
0,7
FORMULA
a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} c(i) * c(j) * c(n-i-j) * j, where c = A010051.
a(n) = A355199(n) - A355196(n) - A355197(n).
EXAMPLE
a(9) = 5; since 9 can be written as the sum of 3 primes in two ways: 2+2+5 = 3+3+3 and the sum of the smallest parts of these partitions is 2+3 = 5, we have a(9) = 5.
MATHEMATICA
Table[Sum[Sum[j (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