A131749 Triangle of successive absolute differences of semiprimes.

4, 2, 6, 1, 3, 9, 1, 2, 1, 10, 0, 1, 3, 4, 14, 1, 1, 0, 3, 1, 15, 0, 1, 2, 2, 5, 6, 21, 1, 1, 0, 2, 0, 5, 1, 22, 1, 0, 1, 1, 3, 3, 2, 3, 25, 1, 0, 0, 1, 0, 3, 0, 2, 1, 26, 1, 0, 0, 0, 1, 1, 4, 4, 6, 7, 33, 1, 0, 0, 0, 0, 1, 0, 4, 0, 6, 1, 34, 0, 1, 1, 1, 1, 1, 2, 2, 6, 6, 0, 1, 35

Offset: 1

Author

Jonathan Vos Post, Oct 23 2007

Keywords

• easy
• nonn
• tabl

Comments

Semiprime analog of A036262. The conjecture analogous to Gilbreath's conjecture is that the leading term (after the second row) is always 0 or 1. First diagonal is semiprimes (A001358). Second diagonal is first differences of semiprimes (A065516).

Examples

			Table begins:
4  6  9 10 14 15 21 22 25 26 33 34 35 38 39 46 49 51 55 57 58 62 65 69 74 77 82 85
2  3  1  4  1  6  1  3  1  7  1  1  3  1  7  3  2  4  2  1  4  3  4  5  3  5  3
1  2  3  3  5  5  2  2  6  6  0  2  2  6  4  1  2  2  1  3  1  1  1  2  2  2
1  1  0  2  0  3  0  4  0  6  2  0  4  2  3  1  0  1  2  2  0  0  1  0  0
0  1  2  2  3  3  4  4  6  4  2  4  2  1  2  1  1  1  0  2  0  1  1  0
1  1  0  1  0  1  0  2  2  2  2  2  1  1  1  0  0  1  2  2  1  0  1
0  1  1  1  1  1  2  0  0  0  0  1  0  0  1  0  1  1  0  1  1  1
1  0  0  0  0  1  2  0  0  0  1  1  0  1  1  1  0  1  1  0  0
1  0  0  0  1  1  2  0  0  1  0  1  1  0  0  1  1  0  1  0
1  0  0  1  0  1  2  0  1  1  1  0  1  0  1  0  1  1  1
1  0  1  1  1  1  2  1  0  0  1  1  1  1  1  1  0  0
1  1  0  0  0  1  1  1  0  1  0  0  0  0  0  1  0
0  1  0  0  1  0  0  1  1  1  0  0  0  0  1  1
1  1  0  1  1  0  1  0  0  1  0  0  0  1  0
0  1  1  0  1  1  1  0  1  1  0  0  1  1
1  0  1  1  0  0  1  1  0  1  0  1  0
1  1  0  1  0  1  0  1  1  1  1  1
0  1  1  1  1  1  1  0  0  0  0
1  0  0  0  0  0  1  0  0  0
1  0  0  0  0  1  1  0  0
1  0  0  0  1  0  1  0
1  0  0  1  1  1  1
1  0  1  0  0  0
1  1  1  0  0
0  0  1  0
0  1  1
1  0
1
etc.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..10011
• Index entries for sequences related to Gilbreath conjecture and transform

Crossrefs

Cf. A001358, A036262, A065516.

Programs

Mathematica

SemiPrimePi[n_] := Sum[ PrimePi[n/Prime[i]] - i + 1, {i, PrimePi[ Sqrt[n]]}]; SemiPrime[n_] := Block[{e = Floor[Log[2, n] + 1], a, b}, a = 2^e; Do[b = 2^p; While[SemiPrimePi@a < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; t[0, n_] := SemiPrime[n]; t[r_, c_] := Abs[t[r - 1, c] - t[r - 1, c + 1]]; Table[t[r - c, c], {r, 13}, {c, r}] // Flatten
(* to construct the table as shown *) mx = 13; Table[t[r, c], {r, 0, mx - 1}, {c, mx - r}] // TableForm (* Robert G. Wilson v, Jun 13 2018 *)