A095195 T(n,0) = prime(n), T(n,k) = T(n,k-1)-T(n-1,k-1), 0<=k<n, triangle read by rows.

2, 3, 1, 5, 2, 1, 7, 2, 0, -1, 11, 4, 2, 2, 3, 13, 2, -2, -4, -6, -9, 17, 4, 2, 4, 8, 14, 23, 19, 2, -2, -4, -8, -16, -30, -53, 23, 4, 2, 4, 8, 16, 32, 62, 115, 29, 6, 2, 0, -4, -12, -28, -60, -122, -237, 31, 2, -4, -6, -6, -2, 10, 38, 98, 220, 457, 37, 6, 4, 8, 14, 20, 22, 12

Offset

1,1

Comments

T(n,0)=A000040(n); T(n,1)=A001223(n-1) for n>1; T(n,2)=A036263(n-2) for n>2; T(n,n-1)=A007442(n) for n>1.

Row k of the array (not the triangle) is the k-th differences of the prime numbers. - Gus Wiseman, Jan 11 2025

Links

Alois P. Heinz, Rows n = 1..141, flattened

Example

Triangle begins:
   2;
   3,  1;
   5,  2,  1;
   7,  2,  0, -1;
  11,  4,  2,  2,  3;
  13,  2, -2, -4, -6, -9;
Alternative: array form read by antidiagonals:
     2,   3,   5,   7,  11,  13,  17,  19,  23,  29,  31,...
     1,   2,   2,   4,   2,   4,   2,   4,   6,   2,   6,...
     1,   0,   2,  -2,   2,  -2,   2,   2,  -4,   4,  -2,...
    -1,   2,  -4,   4,  -4,   4,   0,  -6,   8,  -6,   0,...
     3,  -6,   8,  -8,   8,  -4,  -6,  14, -14,   6,   4,...
    -9,  14, -16,  16, -12,  -2,  20, -28,  20,  -2,  -8,...
    23, -30,  32, -28,  10,  22, -48,  48, -22,  -6,  10,..,
   -53,  62, -60,  38,  12, -70,  96, -70,  16,  16, -12,...
   115,-122,  98, -26, -82, 166,-166,  86,   0, -28,  28,...
  -237, 220,-124, -56, 248,-332, 252, -86, -28,  56, -98,...
   457,-344,  68, 304,-580, 584,-338,  58,  84,-154, 308,...

Maple

A095195A := proc(n, k) # array, k>=0, n>=0
    option remember;
    if n =0 then
        ithprime(k+1) ;
    else
        procname(n-1, k+1)-procname(n-1, k) ;
    end if;
end proc:
A095195 := proc(n, k) # triangle, 0<=k<n, n>=1
        A095195A(k, n-k-1) ;
end proc: # R. J. Mathar, Sep 19 2013

Mathematica

T[n_, 0] := Prime[n]; T[n_, k_] /; 0 <= k < n := T[n, k] = T[n, k-1] - T[n-1, k-1]; Table[T[n, k], {n, 1, 12}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Feb 01 2017 *)
nn=6;
t=Table[Differences[Prime[Range[nn]], k], {k, 0, nn}];
Table[t[[j, i-j+1]], {i, nn}, {j, i}] (* Gus Wiseman, Jan 11 2025 *)

Prog

(Haskell)
a095195 n k = a095195_tabl !! (n-1) !! (k-1)
a095195_row n = a095195_tabl !! (n-1)
a095195_tabl = f a000040_list [] where
   f (p:ps) xs = ys : f ps ys where ys = scanl (-) p xs
-- Reinhard Zumkeller, Oct 10 2013

Crossrefs

Cf. A036262-A036271.

Cf. A140119 (row sums).

Below, the inclusive primes (A008578) are 1 followed by A000040. See also A075526.

Rows of the array (columns of the triangle) begin: A000040, A001223, A036263.

Column n = 1 of the array is A007442, inclusive A030016.

The version for partition numbers is A175804, see A053445, A281425, A320590.

First position of 0 is A376678, inclusive A376855.

Absolute antidiagonal-sums are A376681, inclusive A376684.

The inclusive version is A376682.

For composite instead of prime we have A377033, see A377034-A377037.

For squarefree instead of prime we have A377038, nonsquarefree A377046.

Column n = 2 of the array is A379542.

Cf. A002808, A064113, A065890, A073783, A258026, A293467, A333254, A376683, A377051.

Sequence in context: A366154 A214055 A066909 * A372640 A229961 A189074

Adjacent sequences: A095192 A095193 A095194 * A095196 A095197 A095198

Keyword

sign,tabl,look,changed

Author

Reinhard Zumkeller, Jun 22 2004

Status

approved