A212212 -id:A212212 - cp's OEIS frontend

A060208 a(n) = 2pi(n) - pi(2n), where pi(i) = A000720(i).

-1, 0, 1, 0, 2, 1, 2, 2, 1, 0, 2, 1, 3, 3, 2, 1, 3, 3, 4, 4, 3, 2, 4, 3, 3, 3, 2, 2, 4, 3, 4, 4, 4, 3, 3, 2, 3, 3, 3, 2, 4, 3, 5, 5, 4, 4, 6, 6, 5, 5, 4, 3, 5, 4, 3, 3, 2, 2, 4, 4, 6, 6, 6, 5, 5, 4, 6, 6, 5, 4, 6, 6, 8, 8, 7, 6, 6, 6, 7, 7, 7, 6, 8, 7, 7, 7, 6, 6, 8, 7, 6, 6, 6, 6, 6, 5, 6, 6, 5, 4, 6, 6, 8, 8, 8, 7, 9, 9, 11, 11, 11, 10, 12, 11, 10, 10, 9, 9, 9, 8, 7, 7

Offset: 1

Labos Elemer, Mar 19 2001

sign

Rosser & Schoenfeld show 2*pi(x) > pi(2*x) for x > 10. - N. J. A. Sloane, Jul 03 2013, corrected Jul 09 2015

			n=100, pi(100)=25, pi(200)=46, 2pi(100)-pi(2*100) =4=a(100)

J. Barkley Rosser and Lowell Schoenfeld, Abstracts of Scientific Communications, Internat. Congress Math., Moscow, 1966, Section 3, Theory of Numbers.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VII.5, p. 235.
Sanford Segal, On Pi(x+y)<=Pi(x)+Pi(y). Transactions American Mathematical Society, 104 (1962), 523-527.

T. D. Noe, Table of n, a(n) for n = 1..10000
Eugene Ehrhart, On prime numbers, Fibonacci Quarterly 26:3 (1988), pp. 271-274. Shows a(n)>0 for n>10.
E. Labos, Illustration

Cf. A000720, A007097, A033844, A060207, A212210, A212211, A212212, A212213.

[2*#PrimesUpTo(n) -#PrimesUpTo(2*n): n in [1..200]]; // G. C. Greubel, Aug 01 2024

f[n_] := 2 PrimePi[n] - PrimePi[2 n]; Array[f, 122] (* Robert G. Wilson v, Aug 12 2011 *)

a(n)=2*primepi(n)-primepi(2*n) \\ Charles R Greathouse IV, Jul 02 2013

[2*prime_pi(n) -prime_pi(2*n) for n in range(1,201)] # G. C. Greubel, Aug 01 2024

a(n) = Mod[2*PrimePi[n], PrimePi[2n]] = 2*A000720(n) - A000720(2n) for n>1.
a(n) ~ 2n log 2 / (log n)^2, by the prime number theorem. - N. J. A. Sloane, Mar 12 2007
a(n) = -A047886(n,n) (see A212210 to A212213). - Reinhard Zumkeller, Apr 15 2008

Edited by N. J. A. Sloane, Jul 03 2013

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

Original entry on oeis.org

-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 *)

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

A060208 a(n) = 2*pi(n) - pi(2*n), where pi(i) = A000720(i).

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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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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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Haskell

Mathematica

A060208 a(n) = 2pi(n) - pi(2n), where pi(i) = A000720(i).