A047885 -id:A047885 - cp's OEIS frontend

A212210 Triangle read by rows: T(n,k) = pi(n) + pi(k) - pi(n+k), n >= 1, 1 <= k <= n, where pi() = A000720().

-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

Offset: 1

N. J. A. Sloane, May 04 2012

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]

			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
  ...

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VII.5, p. 235.

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).

Cf. A000720, A212211, A212212, A212213, A060208, A047885, A047886.

Left diagonal is -A010051.

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

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 *)

A212213 Array read by antidiagonals: pi(n) + pi(k) - pi(n+k), where pi() = A000720.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 2, 1, 2, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 2, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0

Offset: 2

N. J. A. Sloane, May 04 2012

It is conjectured that pi(x) + pi(y) >= pi(x+y) for 1 < y <= x.

			Array begins:
  0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, ...
  0, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, ...
  0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, ...
  0, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, ...
  0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, ...
  1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, ...
  ...

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VII.5, p. 235.

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
P. Erdős 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.

Cf. A000720, A047885, A047886, A060208, A212210-A212213.

t[n_, k_] := PrimePi[n] + PrimePi[k] - PrimePi[n + k]; Table[t[n - k + 2, k], {n, 0, 15}, {k, 2, n}] // Flatten (* Jean-François Alcover, Dec 31 2012 *)

A047886 Triangle read by rows: T(n,k) = pi(n+k) - pi(n) - pi(k), where pi() = A000720 (n >= 0, 0 <= k <= n).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, -1, -1, -2, 0, 1, 0, -1, -1, -1, -1, 0, 0, -1, -2, -1, -2, -1, -2, 0, 0, -1, -1, -1, -1, -1, -2, -2, 0, 0, 0, -1, 0, -1, -1, -2, -1, -1, 0, 1, 0, 0, 0, -1, -1, -1, -1, 0, 0, 0, 0, 0, -1, -1, -2, -1, -2, -1, -1, -1, -2, 0, 1, 0, -1

Offset: 0

N. J. A. Sloane

T(n,0)=0; for n > 0: T(n,1)=A010051(n); T(n,n)=-A060208(n). - Reinhard Zumkeller, Apr 15 2008

A212210-A212213 are the preferred versions of this array.

			Triangle begins
  0;
  0,  1;
  0,  1,  0;
  0,  0,  0, -1;
  0,  1,  0,  0,  0;
  0,  0,  0, -1, -1, -2;
  ...

Cf. A047885, A212210-A212213.

Flatten[Table[PrimePi[n+k]-PrimePi[n]-PrimePi[k],{n,0,20},{k,0,n}]] (* Harvey P. Dale, Feb 22 2012 *)

More terms from James Sellers, Dec 22 1999

A212212 Array read by antidiagonals: pi(n) + pi(k) - pi(n+k), where pi() = A000720.

Original entry on oeis.org

-1, -1, -1, 0, 0, 0, -1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 2, 1, 2, 1, 2, 1, 0, -1, 0, 1, 1, 1, 1, 1, 1, 0, -1, 0, 0, 1, 1, 2, 1, 2, 1, 1, 0, 0, -1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, -1, 0, 0, 1, 0, 1, 1, 2, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 0

Offset: 1

N. J. A. Sloane, May 04 2012

It is conjectured that pi(x) + pi(y) >= pi(x+y) for 1 < y <= x.

			Array begins:
  -1, -1,  0, -1,  0, -1,  0,  0,  0, -1,  0, -1, ...
  -1,  0,  0,  0,  0,  0,  1,  1,  0,  0,  0,  0, ...
   0,  0,  1,  0,  1,  1,  2,  1,  1,  0,  1,  1, ...
  -1,  0,  0,  0,  1,  1,  1,  1,  0,  0,  1,  1, ...
   0,  0,  1,  1,  2,  1,  2,  1,  1,  1,  2,  1, ...
  -1,  0,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
   0,  1,  2,  1,  2,  1,  2,  2,  2,  1,  2,  1, ...
   0,  1,  1,  1,  1,  1,  2,  2,  1,  1,  1,  1, ...
   ...

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VII.5, p. 235.

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
P. Erdős 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).

Cf. A000720, A212210-A212213, A060208, A047885, A047886. First row and column are -A010051.

a[n_, k_] := PrimePi[n] + PrimePi[k] - PrimePi[n+k]; Flatten[ Table[a[n-k, k], {n, 1, 15}, {k, 1, n-1}]] (* Jean-François Alcover, Jul 18 2012 *)

A212211 Triangle read by rows: T(n,k) = pi(n) + pi(k) - pi(n+k), n >= 2, 2 <= k <= n, where pi() = A000720().

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 1, 1, 2, 0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 0, 1, 0, 1, 1, 2, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 2, 1, 2, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3

Offset: 2

N. J. A. Sloane, May 04 2012

It is conjectured that pi(x) + pi(y) >= pi(x+y) for 1 < y <= x.

			Triangle begins:
  0,
  0, 1,
  0, 0, 0,
  0, 1, 1, 2,
  0, 1, 1, 1, 1,
  1, 2, 1, 2, 1, 2,
  1, 1, 1, 1, 1, 2, 2,
  0, 1, 0, 1, 1, 2, 1, 1,
  0, 0, 0, 1, 1, 1, 1, 0, 0,
  0, 1, 1, 2, 1, 2, 1, 1, 1, 2,
  0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
  ...

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VII.5, p. 235.

Reinhard Zumkeller, Rows n = 2..150 of triangle, flattened
P. Erdős 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).

Cf. A000720, A212210, A212212, A212213, A060208, A047885, A047886.

a212211 n k = a212211_tabl !! (n-2) !! (k-2)
a212211_tabl = map a212211_row [2..]
a212211_row n = zipWith (-)
   (map (+ a000720 n) $ take (n - 1) $ tail a000720_list)
   (drop (n + 1) a000720_list)
-- Reinhard Zumkeller, May 04 2012

t[n_, k_] := PrimePi[n] + PrimePi[k] - PrimePi[n+k]; Flatten[ Table[t[n, k], {n, 2, 13}, {k, 2, n}]] (* Jean-François Alcover, May 21 2012 *)

cp's OEIS Frontend

A212210 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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A212213 Array read by antidiagonals: pi(n) + pi(k) - pi(n+k), where pi() = A000720.

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A047886 Triangle read by rows: T(n,k) = pi(n+k) - pi(n) - pi(k), where pi() = A000720 (n >= 0, 0 <= k <= n).

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A212212 Array read by antidiagonals: pi(n) + pi(k) - pi(n+k), where pi() = A000720.

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A212211 Triangle read by rows: T(n,k) = pi(n) + pi(k) - pi(n+k), n >= 2, 2 <= k <= n, where pi() = A000720().

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Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica