A194939 Table T read by rows, where T(n, k) is the sum of the largest k primes up to and including prime(n), for 1 <= k <= n.

2, 3, 5, 5, 8, 10, 7, 12, 15, 17, 11, 18, 23, 26, 28, 13, 24, 31, 36, 39, 41, 17, 30, 41, 48, 53, 56, 58, 19, 36, 49, 60, 67, 72, 75, 77, 23, 42, 59, 72, 83, 90, 95, 98, 100, 29, 52, 71, 88, 101, 112, 119, 124, 127, 129, 31, 60, 83, 102, 119, 132, 143, 150, 155, 158, 160

Offset: 1

Author

Alonso del Arte, Sep 07 2011

Keywords

• nonn
• easy
• tabl

Comments

From the left, the second column gives the sums of two consecutive primes, the third column gives the sums of three consecutive primes, etc. Thus, from the right, the rightmost column gives the running sum of all prime numbers up to that row.

This triangle is the mirror image of A143121: left border are the primes (right border in the other one) while the right border is the sum of the first n primes (A007504, left border in the other one). Row sums are given by A014285, just like the other triangle.

On odd numbered rows, the central entry is exactly the same as the corresponding position in A143121: T(n, (n + 1)/2) = A143121(n, (n + 1)/2). The rest of the row is of course the reverse.

Examples

			First few rows of triangle are:
2
3,   5
5,   8, 10
7,  12, 15, 17
11, 18, 23, 26, 28
...
T(5, 2) = 18 because the sum of the fourth and fifth primes (two consecutive primes) is 7 + 11 = 18.
T(5, 3) = 23 because the sum of the third, fourth and fifth primes (three consecutive primes) is 5 + 7 + 11 = 23.
		

Links

• Michel Marcus, Table of n, a(n) for n = 1..5050 (first 100 rows)

Crossrefs

Cf. A143121 (rows reversed), A014285 (row sums).

Cf. A000040 (column k=1), A007504 (main diagonal).

Cf. A067377.

Programs

Mathematica

a[n_, k_] := a[n, k] = Plus@@Prime[Range[n - k + 1, n]]; Column[Table[a[n, k], {n, 15}, {k, n}], Center]

Formula

T(n, k) = Sum_{i = n-k+1..n} prime(i), where prime(i) is the i-th prime number.

Extensions

More terms from Michel Marcus, Aug 31 2020

New name from David A. Corneth, Aug 31 2020