A073882 -id:A073882 - cp's OEIS frontend

A038107 Number of primes < n^2.

0, 0, 2, 4, 6, 9, 11, 15, 18, 22, 25, 30, 34, 39, 44, 48, 54, 61, 66, 72, 78, 85, 92, 99, 105, 114, 122, 129, 137, 146, 154, 162, 172, 181, 191, 200, 210, 219, 228, 240, 251, 263, 274, 283, 295, 306, 319, 329, 342, 357, 367, 378, 393, 409, 421, 434, 445, 457, 474

Offset: 0

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

nonn

Also number of primes <= n^2 since n^2 is not prime.
Also the number of primes contained within an n X n square spiral. - William A. Tedeschi, Mar 03 2008
For large n, these numbers closely approximate the sum of primes less than n. For example, n = 10^10, sum of primes < n = 2220822432581729238. The number of primes < (10^10)^2 = 10^20 = 2220819602560918840. The error is 0.0000012743... The derivation of this is in the link Sum of Primes. - Cino Hilliard, Jun 09 2008
a(n) - A000720(n) = A073882(n) - A010051(n) = A117490(n). - Reinhard Zumkeller, May 20 2010
A061265(a(n)) = 1 for n > 1. - Reinhard Zumkeller, Apr 15 2013
From Zhi-Wei Sun, Feb 17 2014: (Start)
Conjecture:
(i) The sequence a(n)^(1/n) (n = 3, 4, ...) is strictly decreasing (to the limit 1).
(ii) If n > 0 is not among 25, 35, 44, 46, 105, then the interval [a(n), a(n+1)] contains at least one prime. (End)
A classical conjecture of Legendre asserts that a(n) < a(n+1) for all n > 0.
Conjecture: All the numbers Sum_{i=j,...,k} 1/a(i) with 1 < j <= k have pairwise distinct fractional parts. - Zhi-Wei Sun, Sep 24 2015

			a(2)=2 because the only primes < 4 are 2 and 3.

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. (See Conjectures 2.14-2.16.)

T. D. Noe, Table of n, a(n) for n = 0..1000
Cino Hilliard, Sum of Primes.
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.
Wikipedia, Legendre's conjecture.

Cf. A014085 (first differences), A111208, A194189, A262408, A262443, A262447, A262462.

a038107 0 = 0
a038107 n = a000720 $ a000290 n
-- Reinhard Zumkeller, Apr 15 2013, Nov 01 2011

A038107 := proc(n) numtheory[pi]( n^2) ; end: seq(A038107(n),n=0..100) ; # R. J. Mathar, Jun 22 2009

Table[PrimePi[n^2], {n, 0, 100}] (* Ray Chandler, Oct 22 2005 *)

a(n)=primepi(n^2) \\ Charles R Greathouse IV, Apr 26 2012

[prime_pi(n^2) for n in range(0, 59)] # Zerinvary Lajos, Jun 06 2009

a(n) = A000720(A000290(n)).

a(n) ~ 1/2 * n^2/log n. - Charles R Greathouse IV, Apr 26 2012

Extended by Ray Chandler, Oct 22 2005

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

A220492 Number of primes p between quarter-squares, Q(n) < p <= Q(n+1), where Q(n) = A002620(n).

Original entry on oeis.org

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

Offset: 0

Omar E. Pol, Feb 04 2013

nonn

It appears that a(n) > 0, if n > 1.
Apparently the above comment is equivalent to the Oppermann's conjecture. - Omar E. Pol, Oct 26 2013
For n > 0, also the number of primes per quarter revolution of the Ulam Spiral. The conjecture implies that there is at least one prime in every turn after the first. - Ruud H.G. van Tol, Jan 30 2024

			When the nonnegative integers are written as an irregular triangle in which the right border gives the quarter-squares without repetitions, a(n) is the number of primes in the n-th row of triangle. See below (note that the prime numbers are in parenthesis):
---------------------------------------
Triangle                          a(n)
---------------------------------------
0;                                 0
1;                                 0
(2);                               1
(3),   4;                          1
(5),   6;                          1
(7),   8,   9;                     1
10,  (11), 12;                     1
(13), 14,  15,   16;               1
(17), 18, (19),  20;               2
21,   22, (23),  24,  25;          1
26,   27,  28,  (29), 30;          1
...

Ruud H.G. van Tol, Table of n, a(n) for n = 0..10000
Wikipedia, Oppermann's conjecture

Partial sums give A220506.

Cf. A000040, A002620, A001477, A014085, A066888, A073882, A222030.

a(n) = #primes([n^2/4, (n+1)^2/4]); \\ Ruud H.G. van Tol, Feb 01 2024

A109819 Product of primes between n and n^2.

Original entry on oeis.org

1, 6, 105, 5005, 37182145, 6685349671, 20496326086283047, 558516101711461766587, 15322117939717490037614688353, 10978895066407230594062391177770267, 150524069716274322800691458531160297587022319, 4335344870389259769484516507207766258600144871515339

Offset: 1

Rick L. Shepherd, Jul 02 2005

			a(3) = 105 because 3, 5 and 7 are the A073882(3) = 3 primes in the interval from 3 to 3^2 inclusive and 3 * 5 * 7 = 105.

Alois P. Heinz, Table of n, a(n) for n = 1..48

Cf. A109818 (sum of same primes), A073882 (number of primes between n and n^2).

Join[{1},Table[Product[Prime[i],{i,If[PrimeQ[n],PrimePi[n],PrimePi[n]+1],PrimePi[n^2]}],{n,2,10}]] (* James C. McMahon, Apr 01 2024 *)

for(n=1,15,print1(prod(k=n,n^2,if(isprime(k),k,1)),","))

log(a(n)) ~ n^2 by the Prime Number Theorem. - Charles R Greathouse IV, Feb 05 2025

More terms from Michel Marcus, Apr 02 2024

A098598 Number of primes in sequences formed from the t digits of n where the latter terms are given by rule b(i)=sum of t previous terms; primes are counted from initial t digits up to the largest term < n^2.

Original entry on oeis.org

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

Offset: 10

Jason Earls, Sep 17 2004

			a(21)=7 because there are 7 primes in the sequence
2,1,3,4,7,11,18,29,47,76,123,199,322.

Cf. A007629, A073882.

A109818 Sum of primes between n and n^2.

Original entry on oeis.org

0, 5, 15, 36, 95, 150, 318, 484, 774, 1043, 1576, 2099, 2886, 3790, 4620, 6040, 7941, 9465, 11541, 13810, 16763, 19982, 23515, 26840, 32253, 37461, 42368, 48394, 55737, 62668, 70112, 80029, 89512, 100678, 111427, 124051, 135954, 148630, 166354

Offset: 1

Rick L. Shepherd, Jul 02 2005

			a(3) = 15 because 3, 5 and 7 are the A073882(3) = 3 primes in the interval from 3 to 3^2 inclusive and 3 + 5 + 7 = 15.

Cf. A109819 (product of same primes), A073882 (number of primes between n and n^2).

Join[{0},Table[Sum[Prime[i],{i,If[PrimeQ[n],PrimePi[n],PrimePi[n]+1],PrimePi[n^2]}],{n,2,39}]] (* James C. McMahon, Apr 02 2024 *)

for(n=1,50,print1(sum(k=n,n^2,if(isprime(k),k)),","))

cp's OEIS Frontend

A038107 Number of primes < n^2.

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

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A220492 Number of primes p between quarter-squares, Q(n) < p <= Q(n+1), where Q(n) = A002620(n).

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A109819 Product of primes between n and n^2.

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A098598 Number of primes in sequences formed from the t digits of n where the latter terms are given by rule b(i)=sum of t previous terms; primes are counted from initial t digits up to the largest term < n^2.

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A109818 Sum of primes between n and n^2.

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PARI