A335188 Total surface area of all rectangular prisms with dimensions p X q X q, where p and q are prime, n = p+q and p<q.

0, 0, 0, 0, 42, 0, 90, 110, 154, 182, 0, 238, 330, 374, 442, 956, 0, 1148, 714, 1484, 874, 1868, 0, 3066, 1242, 2588, 0, 2844, 0, 4722, 1914, 3740, 2170, 6626, 0, 9304, 0, 2790, 3034, 8762, 0, 12760, 3690, 11162, 4042, 12050, 0, 21350, 4794, 18824, 0, 14874, 0, 26846, 6042, 17738

Offset

1,5

Comments

Consider the Goldbach partitions of n into two distinct parts (p,q) with p < q. Then a(n) is the sum of all the products, 2q*(2p+q), using the corresponding parts from each (p,q) pair.

Total area of all rectangles with dimensions (2q) X (2p+q), with p and q prime, n = p+q and p < q.

Links

Table of n, a(n) for n=1..56.

Eric Weisstein's World of Mathematics, Goldbach Partition

Wikipedia, Goldbach's conjecture

Index entries for sequences related to Goldbach conjecture

Index entries for sequences related to partitions

Formula

a(n) = 2 * Sum_{i=1..floor((n-1)/2)} (n-i) * (n+i) * c(i) * c(n-i), where c is the prime characteristic (A010051).

Example

a(6) = 0; no prisms are possible.
a(7) = 90; 1 prism with surface area = 2*(2*5 + 5*5 + 2*5) = 90.
a(8) = 110; 1 prism with surface area = 2*(3*5 + 5*5 + 3*5) = 110.
a(9) = 154; 1 prism with surface area = 2*(2*7 + 7*7 + 2*7) = 154.

Mathematica

Table[2 Sum[(n - i)*(n + i)*(PrimePi[i] - PrimePi[i - 1])*(PrimePi[n - i] - PrimePi[n - i - 1]), {i, Floor[(n - 1)/2]}], {n, 80}]

Crossrefs

Cf. A010051.

Sequence in context: A023932 A243831 A022074 * A263303 A263288 A216799

Adjacent sequences: A335185 A335186 A335187 * A335189 A335190 A335191

Keyword

nonn,easy

Author

Wesley Ivan Hurt, May 25 2020

Status

approved