A060208 -id:A060208 - 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 *)

A060207 Start at 2^n, iterate function PrimePi (A000720) until fixed point is reached; sequence gives number of steps.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 21, 22, 22, 22, 22, 22

Offset: 0

Labos Elemer, Mar 19 2001

nonn

A007097(a(n) - 2) <= 2^n < A007097(a(n) - 1). - David Wasserman, May 31 2002

			n=24, the relevant list is: {16777216,1077871,84115,8198,1028,172,39,12,5,3,2,1,0}, its length a(24)=13.

S. Segal, On pi(x+y)<=pi(x)+pi(y), Transactions American Mathematical Society, 104 (1962), 523-527.

Cf. A060208, A007097, A000720, A033844, A071682.

Table[Length[FixedPointList[PrimePi, 2^w]]-1, {w, 0, 32}]
f[n_] := Length@ NestWhileList[ PrimePi, 2^n, # > 0 &]; Array[f, 48, 0] (* Robert G. Wilson v, Aug 12 2011 *)

a(n) = {my(c=2, k=2^n); while(k=primepi(k), c++); c; } \\ Jinyuan Wang, May 16 2020

More terms from David Wasserman, May 31 2002

A060303 Number of primes below n^2 does not exceed n times the number of primes below n.

Original entry on oeis.org

0, 0, 2, 2, 6, 7, 13, 14, 14, 15, 25, 26, 39, 40, 42, 42, 58, 60, 80, 82, 83, 84, 108, 111, 111, 112, 114, 115, 144, 146, 179, 180, 182, 183, 185, 186, 225, 228, 228, 229, 270, 272, 319, 321, 324, 325, 376, 378, 378, 383, 387, 387, 439, 443, 446, 451, 455, 454

Offset: 1

Labos Elemer, Mar 26 2001

nonn

			pi(100) = 25, 10*pi(10) = 40, a(10) = 40-25 = 15.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Sanford L. Segal, On Pi(x+y)<=Pi(x)+Pi(y), Transactions American Mathematical Society, Vol. 104, No. 3 (1962), pp. 523-527.

Cf. A000720, A060199, A060208.

Table[n*PrimePi[n]-PrimePi[n^2], {n, 1, 100}]

Offset corrected by Amiram Eldar, Sep 06 2024

A060304 Number of primes below n^3 does not exceed n times the number of primes below n^2.

Original entry on oeis.org

0, 0, 3, 6, 15, 19, 37, 47, 69, 82, 113, 139, 180, 216, 244, 300, 381, 423, 486, 553, 638, 726, 820, 887, 1029, 1152, 1256, 1376, 1527, 1659, 1794, 1992, 2156, 2357, 2517, 2739, 2909, 3085, 3365, 3627, 3933, 4200, 4380, 4687, 4960, 5313, 5547, 5917, 6395

Offset: 0

Labos Elemer, Mar 26 2001

nonn

			n=10, 10*pi(100)=250, pi(1000)=168, a(10)=250-168=82.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. Segal, On π(x+y)<=π(x)+π(y), Transactions American Mathematical Society, 104 (1962), 523-527.

Cf. A060199, A060208, A000720.

Table[n*PrimePi[n^2]-PrimePi[n^3], {n, 1, 100}]

a(n) = n*pi(n*n) - pi(n*n*n). - Jonathan Sondow, Feb 17 2014

a(n) = n*A038107(n) - A038098(n). - Michel Marcus, Feb 17 2014

A259922 a(n)= Sum_{2 < prime p <= n} c_p - Sum_{n < prime p < 2*n} c_p, where 2^c_p is the greatest power of 2 dividing p-1.

Original entry on oeis.org

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

Offset: 1

Vladimir Shevelev, Jul 09 2015

sign

It is known that, for n>10, pi(2*n) < 2*pi(n), where pi(n) is the number of primes not exceeding n (A000720). Thus, for n>10, in the interval (1,n] we have more primes than in the interval (n,2*n).

In connection with this, it is natural to conjecture that there exists a number N such that a(n)>0 for all n >= N.

Peter J. C. Moses, Table of n, a(n) for n = 1..2000

Cf. A007814, A060208, A259788, A259897.

Map[Total[Flatten[Map[IntegerExponent[Select[#,PrimeQ]-1,2]&,{Range[3,#],Range[#+1,2#-1]}]{1,-1}]]&,Range[50]]

More terms from Peter J. C. Moses, Jul 09 2015

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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A060207 Start at 2^n, iterate function PrimePi (A000720) until fixed point is reached; sequence gives number of steps.

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PARI

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A060303 Number of primes below n^2 does not exceed n times the number of primes below n.

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A060304 Number of primes below n^3 does not exceed n times the number of primes below n^2.

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