A038107 -id:A038107 - cp's OEIS frontend

A221056 Numbers k such that there is no square between prime(k) and prime(k+1).

1, 3, 5, 7, 8, 10, 12, 13, 14, 16, 17, 19, 20, 21, 23, 24, 26, 27, 28, 29, 31, 32, 33, 35, 36, 37, 38, 40, 41, 42, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 84, 86

Offset: 1

Reinhard Zumkeller, Apr 15 2013

nonn

A061265(a(n)) = 0;

a(n) = A049084(A224363(n)); A000040(a(n)) = A224363(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Legendre's Conjecture
Wikipedia, Legendre's conjecture

Cf. A038107, A014085.

import Data.List (elemIndices)
a221056 n = a221056_list !! (n-1)
a221056_list = map (+ 1) $ elemIndices 0 a061265_list

Select[Range[86], Ceiling[Sqrt[Prime[#]]]^2 > Prime[# + 1] &] (* Zak Seidov, Apr 16 2013 *)

{for (n = 1, 86, ceil (sqrt (prime (n)))^2 > prime (n + 1) && print1 (n ","))} \\ Zak Seidov, Apr 16 2013

A237656 Least positive integer m such that {A000720(k^2): k = 1, ..., m} contains a complete system of residues modulo n, or 0 if such a number m does not exist.

Original entry on oeis.org

1, 5, 3, 6, 8, 10, 18, 17, 30, 41, 28, 43, 29, 33, 43, 27, 66, 47, 98, 105, 155, 114, 113, 100, 49, 62, 118, 146, 85, 125, 80, 117, 74, 101, 167, 137, 168, 282, 176, 287, 129, 178, 151, 140, 163, 139, 262, 267, 277, 234, 285, 188, 203, 163, 192, 239, 188, 241, 252, 252

Offset: 1

Zhi-Wei Sun, Feb 10 2014

nonn

Conjecture: a(n) is always positive. Moreover, a(n) < 2*prime(n+1) - 2 for all n > 0.

Note that a(21) = 155 = 2*prime(22) - 3.

			a(5) = 8 since {A000720(k^2): k = 1, ..., 8} = {0, 2, 4, 6, 9, 11, 15, 18} contains a complete system of residues modulo 5, but {A000720(k^2): k = 1, ..., 7} contains no integer congruent to 3 modulo 5.

Chai Wah Wu, Table of n, a(n) for n = 1..1011 (n = 1..100 from Zhi-Wei Sun)
Zhi-Wei Sun, On a^n+bn modulo m, preprint, arXiv:1312.1166 [math.NT], 2013-2014.

Cf. A000290, A000720, A038107, A237643.

q[m_,n_]:=Length[Union[Table[Mod[PrimePi[k^2],n],{k,1,m}]]]
Do[Do[If[q[m,n]==n,Print[n," ",m];Goto[aa]],{m,n,2*Prime[n+1]-3}];
Print[n," ",0];Label[aa];Continue,{n,1,60}]

A262746 Number of ordered ways to write n as x^2 + y^2 + pi(z^2) with 0 <= x <= y and z > 0 such that 2xy + 3 is prime, where pi(m) denotes the number of primes not exceeding m.

Original entry on oeis.org

1, 2, 1, 3, 2, 3, 2, 3, 3, 3, 4, 2, 3, 2, 3, 3, 3, 3, 4, 5, 1, 4, 4, 3, 3, 6, 5, 2, 4, 4, 6, 3, 2, 5, 6, 3, 1, 6, 4, 4, 4, 4, 4, 4, 4, 2, 6, 4, 3, 7, 5, 5, 4, 6, 5, 7, 2, 3, 8, 3, 5, 3, 4, 6, 7, 5, 4, 7, 4, 6, 7, 3, 4, 8, 7, 4, 3, 4, 4, 11, 3, 4, 9, 4, 4, 6, 7, 2, 9, 6, 3, 6, 4, 6, 7, 3, 5, 8, 5, 5

Offset: 1

Zhi-Wei Sun, Sep 29 2015

nonn

Conjectures:
(i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 3, 21, 37, 117, 184. Also, any integer n > 8 can be written as x^2 + y^2 + pi(z^2), where x, y and z are integers with x+y (or z) odd.
(ii) Each n = 8,9,... can be written as p^2 + pi(x^2) + pi(y^2), where p is prime, and x and y are positive integers.
(iii) Every n = 8,9,... can be written as pi(p^2) + pi(x^2) + pi(y^2), where p is prime, and x and y are positive integers.
Note that pi(x^2) > n if x > n > 0. We have verified that a(n) > 0 for all n = 1,...,10^6.

			a(1) = 1 since 1 = 0^2 + 1^2 + pi(1^2) with 2*0*1 + 3 = 3 prime.
a(2) = 2 since 2 = 0^2 + 0^2 + pi(2^2) = 1^2 + 1^2 + pi(1^2) with 2*0*0 + 3 = 3 and 2*1*1 + 3 = 5 both prime.
a(3) = 1 since 3 = 0^2 + 1^2 + pi(2^2) with 2*0*1 + 3 = 3 prime.
a(21) = 1 since 21 = 1^2 + 4^2 + pi(3^2) with 2*1*4 + 3 = 11 prime.
a(37) = 1 since 37 = 1^2 + 5^2 + pi(6^2) with 2*1*5 + 3 = 13 prime.
a(117) = 1 since 117 = 0^2 + 5^2 + pi(22^2) with 2*0*5 + 3 = 3 prime.
a(184) = 1 since 184 = 0^2 + 13^2 + pi(7^2) with 2*0*13 + 3 = 3 prime.

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A000290, A000720, A001481, A038107, A262408, A262447, A262698, A262731.

pi[n_]:=PrimePi[n^2]
SQ[n_]:=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[pi[z]>n,Goto[aa]];Do[If[SQ[n-pi[z]-y^2]&&PrimeQ[2y*Sqrt[n-pi[z]-y^2]+3],r=r+1],{y,0,Sqrt[(n-pi[z])/2]}];Continue,{z,1,n}];Label[aa];Print[n," ",r];Continue,{n,1,100}]

A262999 Number of ordered pairs (k, m) with k > 0 and m > 0 such that n = pi(k*(k+1)/2) + pi(m^2), where pi(x) denotes the number of primes not exceeding x.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Oct 07 2015

nonn

Conjecture: a(n) > 0 for all n > 1, and a(n) = 1 only for n = 3, 5, 7, 35, 37, 217, 7439, 10381.
We have verified this for n up to 120000.
See also A262995, A263001 and A263020 for similar conjectures.

			a(2) = 2 since 2 = pi(1*2/2) + pi(2^2) = pi(2*3/2) + pi(1^2).
a(3) = 1 since 3 = pi(3*4/2) + pi(1^2).
a(5) = 1 since 5 = pi(3*4/2) + pi(2^2).
a(7) = 1 since 7 = pi(3*4/2) + pi(3^2).
a(35) = 1 since 35 = pi(13*14/2) + pi(6^2).
a(37) = 1 since 37 = pi(3*4/2) + pi(12^2).
a(217) = 1 since 217 = pi(17*18/2) + pi(33^2).
a(590) = 1 since 590 = 58 + 532 = pi(23*24/2) + pi(62^2).
a(7439) = 1 since 7439 = 3854 + 3585 = pi(269*270/2) + pi(183^2).
a(10381) = 1 since 10381 = 1875 + 8506 = pi(179*180/2) + pi(296^2).

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000217, A000290, A000720, A038107, A111208, A262995, A263001, A263020.

s[n_]:=s[n]=PrimePi[n^2]
t[n_]:=t[n]=PrimePi[n(n+1)/2]
Do[r=0;Do[If[s[k]>n,Goto[bb]];Do[If[t[j]>n-s[k],Goto[aa]];If[t[j]==n-s[k],r=r+1];Continue,{j,1,n-s[k]+1}];Label[aa];Continue,{k,1,n}];Label[bb];Print[n," ",r];Continue,{n,1,100}]

A038098 Number of primes < n^3.

Original entry on oeis.org

0, 4, 9, 18, 30, 47, 68, 97, 129, 168, 217, 269, 327, 400, 476, 564, 656, 765, 882, 1007, 1147, 1298, 1457, 1633, 1821, 2020, 2227, 2460, 2707, 2961, 3228, 3512, 3817, 4137, 4483, 4821, 5194, 5579, 5995, 6413, 6850, 7308, 7789, 8293

Offset: 1

Joe K. Crump (joecr(AT)carolina.rr.com)

nonn

From Zhi-Wei Sun, Oct 17 2015: (Start)
Conjecture: (i) For any integer k > 2 the sequence pi(n^k)/n^k (n = 2,3,...) is strictly decreasing, where pi(x) denotes the number of primes not exceeding x.
(ii) All the numbers pi(n^2)/n^2 (n = 1,2,3,...) are pairwise distinct. Moreover, we have pi(n^2)/n^2 > pi((n+1)^2)/(n+1)^2 for all n > 15646.
(End)

			a(2)=4 because the only primes < 8 are 2,3,5 and 7.

R. J. Mathar, Table of n, a(n) for n = 1..500

Cf. A014085, A038107, A060199 (first differences).

vector(100, n, primepi(n^3)) \\ Altug Alkan, Oct 17 2015

[prime_pi(n^3) for n in range(1, 45)] # Zerinvary Lajos, Jun 06 2009

a(n) = A000720(A000578(n)). - Michel Marcus, Sep 02 2013

A117490 Number of primes between n and n^2 (with n and n^2 excluded).

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 11, 14, 18, 21, 25, 29, 33, 38, 42, 48, 54, 59, 64, 70, 77, 84, 90, 96, 105, 113, 120, 128, 136, 144, 151, 161, 170, 180, 189, 199, 207, 216, 228, 239, 250, 261, 269, 281, 292, 305, 314, 327, 342, 352, 363, 378, 393, 405, 418, 429, 441, 458, 470

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Mar 22 2006

A famous Japanese mathematics book states that this sequence is nonzero (for n>1) if the Riemann Hypothesis is true, but this statement seems to be false.

If the n-th prime is denoted by p(n) then a(j) = number of nonzero values of floor (j^2/p(n)), over all n >= 1, (derived from A165974). - Christopher Hunt Gribble, Oct 03 2009

			For n = 5: between 5+1 = 6 and 5^2-1 = 24 there are the following six primes: 7, 11, 13, 17, 19, 23.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000720, A010051, A014085, A038107, A060715, A073882, A117491, A165974.

P:=proc(n) local i,j,np; for i from 1 by 1 to n do np:=0; for j from i+1 by 1 to i^2-1 do if isprime(j) then np:=np+1; fi; od; print(np); od; end: P(100);

a[n_] := PrimePi[n^2 - 1] - PrimePi[n]; Array[a, 59] (* Robert G. Wilson v, Apr 06 2006 *)

a(n) = pi(n^2) - pi(n), cf. A000720.

a(n) = A038107(n) - A000720(n) = A073882(n) - A010051(n). - Reinhard Zumkeller, May 20 2010

A237657 a(n) = |{n < m < 2*n: pi(m) and pi(m^2) are both prime}|, where pi(.) is given by A000720.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 5, 5, 4, 3, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 6, 6, 6, 5, 4, 4, 4, 4, 5, 5, 5, 5, 5, 4, 4

Offset: 1

Zhi-Wei Sun, Feb 10 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 8.
(ii) For any integer n > 1 there is a prime p <= n such that n + pi(p) is prime. Also, for n > 5 there is a prime p with n < p < 2*n such that pi(p) is prime.
(iii) For each n > 20, there is a prime p with n < p < 2*n such that pi(p^2) is prime.

			a(4) = 1 since pi(6) = 3 and pi(6^2) = 11 are both prime.
a(10) = 1 since pi(17) = 7 and pi(17^2) = 61 are both prime.
a(17) = 1 since pi(33) = 11 and pi(33^2) = 181 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A000720, A038107, A237578, A237643, A237656.

q[n_]:=PrimeQ[PrimePi[n]]&&PrimeQ[PrimePi[n^2]]
a[n_]:=Sum[If[q[m],1,0],{m,n+1,2n-1}]
Table[a[n],{n,1,70}]

A278114 Number of primes <= 2n^2.

Original entry on oeis.org

1, 4, 7, 11, 15, 20, 25, 31, 37, 46, 53, 61, 68, 77, 87, 97, 106, 118, 128, 139, 152, 163, 177, 190, 204, 217, 231, 247, 263, 278, 293, 309, 326, 344, 363, 377, 399, 418, 436, 452, 474, 492, 516, 536, 558, 580, 600, 623, 647, 669, 692, 713, 738, 765, 789, 816, 842, 867

Offset: 1

Jason Kimberley, Feb 09 2017

This is the row length sequence for both A278113 and A278115.

Jason Kimberley, Table of n, a(n) for n = 1..5000

Cf. A038107, A278100.

A278114:=func;
[A278114(n):n in[1..58]];

Table[PrimePi[2 n^2], {n, 58}] (* Michael De Vlieger, Feb 17 2017 *)

a(n) = primepi(2*n^2); \\ Michel Marcus, Feb 15 2017

a(n) = A000720(A001105(n)).

A344316 Number of primes appearing along the border of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows.

Original entry on oeis.org

0, 2, 3, 4, 5, 7, 7, 8, 8, 10, 9, 13, 12, 13, 12, 12, 13, 20, 14, 17, 17, 19, 16, 22, 18, 22, 19, 23, 19, 31, 18, 26, 24, 26, 25, 31, 18, 27, 28, 30, 22, 39, 25, 30, 31, 37, 26, 41, 29, 37, 32, 42, 28, 44, 31, 39, 30, 41, 32, 51, 33, 39, 40, 41, 36, 52, 35, 44, 39, 50, 39, 52, 39

Offset: 1

Wesley Ivan Hurt, May 14 2021

nonn

			                                                      [1   2  3  4  5]
                                      [1   2  3  4]   [6   7  8  9 10]
                            [1 2 3]   [5   6  7  8]   [11 12 13 14 15]
                   [1 2]    [4 5 6]   [9  10 11 12]   [16 17 18 19 20]
           [1]     [3 4]    [7 8 9]   [13 14 15 16]   [21 22 23 24 25]
------------------------------------------------------------------------
  n         1        2         3            4                 5
------------------------------------------------------------------------
  a(n)      0        2         3            4                 5
------------------------------------------------------------------------
  primes   {}      {2,3}    {2,3,7}    {2,3,5,13}       {2,3,5,11,23}
------------------------------------------------------------------------

Cf. A000720 (pi), A038107, A221490, A344349.

Table[PrimePi[n] + PrimePi[n^2 - 1] - PrimePi[n*(n - 1)] + Sum[PrimePi[n*k + 1] - PrimePi[n*k], {k, n - 2}], {n, 100}]

a(n) = pi(n) + pi(n^2-1) - pi(n^2-n) + Sum_{k=1..n-2} (pi(n*k+1) - pi(n*k)).

A078435 Number of composites <= n^2.

Original entry on oeis.org

1, 2, 5, 10, 16, 25, 34, 46, 59, 75, 91, 110, 130, 152, 177, 202, 228, 258, 289, 322, 356, 392, 430, 471, 511, 554, 600, 647, 695, 746, 799, 852, 908, 965, 1025, 1086, 1150, 1216, 1281, 1349, 1418, 1490, 1566, 1641, 1719, 1797, 1880, 1962, 2044, 2133, 2223

Offset: 1

John E. Lenz (jel5010(AT)yahoo.com), Dec 30 2002

nonn

			a(3)=5 because the only composites <= 9 are 1, 4, 6, 8 and 9.

NumComposites := proc(N::posint) local count, i:count := 0:for i from 1 to N do if not isprime(i) then count := count + 1 fi:od: count;end:seq(NumComposites(k^2), k=1..51); # Zerinvary Lajos, May 26 2008
A038107 := proc(n) numtheory[pi]( n^2) ; end: A078435 := proc(n) n^2-A038107(n) ; end: seq(A078435(n),n=1..100) ; # R. J. Mathar, Jun 22 2009

a[n_] := n^2 - PrimePi[n^2];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 20 2024, after R. J. Mathar *)

a(n) = n^2 - A038107(n). - R. J. Mathar, Jun 22 2009

cp's OEIS Frontend

A221056 Numbers k such that there is no square between prime(k) and prime(k+1).

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A237656 Least positive integer m such that {A000720(k^2): k = 1, ..., m} contains a complete system of residues modulo n, or 0 if such a number m does not exist.

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A262746 Number of ordered ways to write n as x^2 + y^2 + pi(z^2) with 0 <= x <= y and z > 0 such that 2*x*y + 3 is prime, where pi(m) denotes the number of primes not exceeding m.

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A262999 Number of ordered pairs (k, m) with k > 0 and m > 0 such that n = pi(k*(k+1)/2) + pi(m^2), where pi(x) denotes the number of primes not exceeding x.

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A038098 Number of primes < n^3.

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A117490 Number of primes between n and n^2 (with n and n^2 excluded).

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A237657 a(n) = |{n < m < 2*n: pi(m) and pi(m^2) are both prime}|, where pi(.) is given by A000720.

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A278114 Number of primes <= 2n^2.

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A262746 Number of ordered ways to write n as x^2 + y^2 + pi(z^2) with 0 <= x <= y and z > 0 such that 2xy + 3 is prime, where pi(m) denotes the number of primes not exceeding m.