A229061 The (n+1)-th term of the n-th differences of the prime sequence.

2, 2, 2, 4, 8, -2, -48, -70, 0, 56, 308, 1014, 798, -2072, -5126, -2820, 434, -1340, 62902, 398032, 1247046, 2834160, 5266626, 7862442, 9510040, 13829302, 37650208, 111410394, 260524940, 468110450, 626899146, 481007522, -490911164, -3217336656, -8570944960

Offset: 0

Author

Jean-François Alcover, Sep 17 2013

Keywords

• sign

Comments

All terms are even. The only zero seems to be a(8), corresponding to A036269(9).

Examples

			The sequences of differences begin:
2,      3,   5,   7,  11,  13,   17,  19,  23,  29, ...
1,      2,   2,   4,   2,   4,    2,   4,   6,   2, ...
1,      0,   2,  -2,   2,  -2,    2,   2,  -4,   4, ...
-1,     2,  -4,   4,  -4,   4,    0,  -6,   8,  -6, ...
3,     -6,   8,  -8,   8,  -4,   -6,  14, -14,   6, ...
-9,    14, -16,  16, -12,  -2,   20, -28,  20,  -2, ...
23,   -30,  32, -28,  10,  22,  -48,  48, -22,  -6, ...
-53,   62, -60,  38,  12, -70,   96, -70,  16,  16, ...
115, -122,  98, -26, -82, 166, -166,  86,   0, -28, ...
etc.
Main diagonal begins:
2, 2, 2, 4, 8, -2, -48, -70, 0, 56, ... .
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 0..98 from Jean-François Alcover)

Crossrefs

Cf. A095195, A036263-A036271, A007442.

Programs

Maple

T:= proc(n, k) option remember;
      `if`(k=0, ithprime(n), T(n+1, k-1)-T(n, k-1))
    end:
a:= n-> T(n+1, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 25 2013

Mathematica

max = 100; row[n_] := Differences[Prime /@ Range[max], n]; Table[row[n], {n, 0, max}] // Diagonal

Formula

a(n) = A095195(2*n+1,n).