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A131749
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Triangle of successive absolute differences of semiprimes.
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2
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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
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OFFSET
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1,1
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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