A143121 Triangle read by rows, T(n,k) = Sum_{j=k..n} prime(j), 1 <= k <= n.

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

Offset: 1

Author

Gary W. Adamson & Roger L. Bagula, Jul 26 2008

Keywords

• nonn
• easy
• tabl

Comments

Left border = A007504, sum of first n primes: (2, 5, 10, 27, 28, 41, ...).

Right border = primes = A000040.

Row sums = A014285: (2, 8, 23, 51, 106, 184, ...).

Examples

			First few rows of the triangle are:
   2;
   5,  3;
  10,  8,  5;
  17, 15, 12,  7;
  28, 26, 23, 18, 11;
  41, 39, 36, 31, 24, 13;
  58, 56, 53, 48, 41, 30, 17;
  ...
T(5,3) = 23 = prime(3) + prime(4) + prime(5) = (5 + 7 + 11).
		

Links

• G. C. Greubel, Rows n=1..100 of triangle, flattened

Crossrefs

Cf. A000040, A007504, A014285, A000012.

Cf. A194939 (rows reversed).

Programs

Magma

[[(&+[NthPrime(j): j in [k..n]]): k in [1..n]]: n in [1..15]]; // G. C. Greubel, Oct 15 2018

Maple

a:=proc(n,k) add(ithprime(j),j=k..n) end: seq(seq(a(n,k),k=1..n),n=1..11); # Muniru A Asiru, Oct 15 2018

Mathematica

a[n_, k_] := a[n, k] = Plus@@Prime[Range[n - k + 1, n]]; Column[Table[a[n, k], {n, 15}, {k, n, 1, -1}], Center] (* Alonso del Arte, Jul 25 2011 *)
Table[Sum[Prime[j], {j, k, n}], {n, 1, 15}, {k, 1, n}]//Flatten (* G. C. Greubel, Oct 15 2018 *)

PARI

a(n,k)=my(s);forprime(p=prime(k),prime(n),s+=p);s \\ Charles R Greathouse IV, Jul 25 2011

Formula

T(n,k) = Sum_{j=k..n} prime(j), 1 <= k <= n, primes = A000040.

Equals A000012 * (A000040 * 0^(n-k)) * A000012.

Extensions

Corrected by Hanke Bremer, Nov 28 2008