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Triangle of successive absolute differences of semiprimes.
2

%I #14 Sep 25 2023 19:26:48

%S 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,

%T 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,

%U 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

%N Triangle of successive absolute differences of semiprimes.

%C 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).

%H Robert G. Wilson v, <a href="/A131749/b131749.txt">Table of n, a(n) for n = 1..10011</a>

%H <a href="/index/Ge#Gilbreath">Index entries for sequences related to Gilbreath conjecture and transform</a>

%e Table begins:

%e 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

%e 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

%e 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

%e 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

%e 0 1 2 2 3 3 4 4 6 4 2 4 2 1 2 1 1 1 0 2 0 1 1 0

%e 1 1 0 1 0 1 0 2 2 2 2 2 1 1 1 0 0 1 2 2 1 0 1

%e 0 1 1 1 1 1 2 0 0 0 0 1 0 0 1 0 1 1 0 1 1 1

%e 1 0 0 0 0 1 2 0 0 0 1 1 0 1 1 1 0 1 1 0 0

%e 1 0 0 0 1 1 2 0 0 1 0 1 1 0 0 1 1 0 1 0

%e 1 0 0 1 0 1 2 0 1 1 1 0 1 0 1 0 1 1 1

%e 1 0 1 1 1 1 2 1 0 0 1 1 1 1 1 1 0 0

%e 1 1 0 0 0 1 1 1 0 1 0 0 0 0 0 1 0

%e 0 1 0 0 1 0 0 1 1 1 0 0 0 0 1 1

%e 1 1 0 1 1 0 1 0 0 1 0 0 0 1 0

%e 0 1 1 0 1 1 1 0 1 1 0 0 1 1

%e 1 0 1 1 0 0 1 1 0 1 0 1 0

%e 1 1 0 1 0 1 0 1 1 1 1 1

%e 0 1 1 1 1 1 1 0 0 0 0

%e 1 0 0 0 0 0 1 0 0 0

%e 1 0 0 0 0 1 1 0 0

%e 1 0 0 0 1 0 1 0

%e 1 0 0 1 1 1 1

%e 1 0 1 0 0 0

%e 1 1 1 0 0

%e 0 0 1 0

%e 0 1 1

%e 1 0

%e 1

%e etc.

%t 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

%t (* 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 *)

%Y Cf. A001358, A036262, A065516.

%K easy,nonn,tabl

%O 1,1

%A _Jonathan Vos Post_, Oct 23 2007