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Sum of the middle parts of the partitions of n into exactly 3 prime parts.
+10
3
0, 0, 0, 0, 0, 0, 2, 2, 3, 5, 3, 5, 8, 8, 5, 12, 10, 17, 8, 15, 12, 26, 10, 24, 19, 33, 23, 48, 23, 57, 16, 54, 31, 80, 16, 71, 36, 93, 42, 112, 26, 122, 31, 120, 48, 145, 23, 148, 48, 160, 59, 190, 42, 218, 39, 196, 71, 240, 35, 246, 62, 252, 108, 310, 53, 366, 59, 340, 96
FORMULA
a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} c(i) * c(j) * c(n-i-j) * i, where c = A010051.
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 middle parts of these partitions is 2+3 = 5, we have a(9) = 5.
MATHEMATICA
Table[Sum[Sum[i (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}]
Sum of the smallest parts of the partitions of n into exactly 3 prime parts.
+10
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
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.
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}]
Sum of all the parts in the partitions of n into exactly 3 prime parts.
+10
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
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.
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}]
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