A194939 -id:A194939 - cp's OEIS frontend

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

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

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, ...).

			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).

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

Cf. A000040, A007504, A014285, A000012.

Cf. A194939 (rows reversed).

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

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

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 *)

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

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

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

Corrected by Hanke Bremer, Nov 28 2008

A221975 Triangle read by rows of the numbers that are the sum of some consecutive Mersenne numbers A000225.

Original entry on oeis.org

1, 3, 4, 7, 10, 11, 15, 22, 25, 26, 31, 46, 53, 56, 57, 63, 94, 109, 116, 119, 120, 127, 190, 221, 236, 243, 246, 247, 255, 382, 445, 476, 491, 498, 501, 502, 511, 766, 893, 956, 987, 1002, 1009, 1012, 1013, 1023, 1534, 1789, 1916, 1979, 2010, 2025, 2032, 2035, 2036

Offset: 1

Omar E. Pol, Feb 12 2013

			Triangle begins:
1;
3,     4;
7,    10,  11;
15,   22,  25,  26;
31,   46,  53,  56,  57;
63,   94, 109, 116, 119, 120;
127, 190, 221, 236, 243, 246, 247;
255, 382, 445, 476, 491, 498, 501, 502;
...

Column 1 is A000225. Right border gives the positive terms of A000295.

Cf. A023758, A141419, A194939.

T(n,k) = sum_{j = n-k+1..n} 2^j - 1, n>=1, k>=1.

T(n,n) = A000295(n+1).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions