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A212210
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Triangle read by rows: T(n,k) = pi(n) + pi(k) - pi(n+k), n >= 1, 1 <= k <= n, where pi() = A000720().
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9
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-1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, 1, 2, -1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 2, 1, 2, 0, 1, 1, 1, 1, 1, 2, 2, 0, 0, 1, 0, 1, 1, 2, 1, 1, -1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 2, 1, 2, 1, 1, 1, 2, -1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 3, 3
(list;
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graph;
refs;
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history;
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OFFSET
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1,15
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COMMENTS
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It is conjectured that pi(x)+pi(y) >= pi(x+y) for 1 < y <= x.
A006093 gives row numbers of rows containing at least one negative term. [Reinhard Zumkeller, May 05 2012]
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REFERENCES
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D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VII.5, p. 235.
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LINKS
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Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened
P. Erdos and J. L. Selfridge, Complete prime subsets of consecutive integers. Proceedings of the Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1971), pp. 1--14. Dept. Comput. Sci., Univ. Manitoba, Winnipeg, Man., 1971. MR0337828 (49 #2597).
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EXAMPLE
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Triangle begins:
-1
-1 0
0 0 1
-1 0 0 0
0 0 1 1 2
-1 0 1 1 1 1
0 1 2 1 2 1 2
0 1 1 1 1 1 2 2
0 0 1 0 1 1 2 1 1
-1 0 0 0 1 1 1 1 0 0
...
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MATHEMATICA
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t[n_, k_] := PrimePi[n] + PrimePi[k] - PrimePi[n + k]; Table[t[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 17 2012 *)
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PROG
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(Haskell)
import Data.List (inits, tails)
a212210 n k = a212210_tabl !! (n-1) !! (k-1)
a212210_row n = a212210_tabl !! (n-1)
a212210_tabl = f $ tail $ zip (inits pis) (tails pis) where
f ((xs, ys) : zss) = (zipWith (-) (map (+ last xs) (xs)) ys) : f zss
pis = a000720_list
-- Reinhard Zumkeller, May 04 2012
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CROSSREFS
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Cf. A000720, A212211, A212212, A212213, A060208, A047885, A047886.
Left diagonal is -A010051.
Sequence in context: A291954 A339885 A106799 * A127499 A198068 A121361
Adjacent sequences: A212207 A212208 A212209 * A212211 A212212 A212213
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KEYWORD
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sign,tabl,nice
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AUTHOR
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N. J. A. Sloane, May 04 2012
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STATUS
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approved
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