A014085 -id:A014085 - cp's OEIS frontend

A076956 Smallest k^2 such that there are exactly n primes between k^2 and (k+1)^2.

1, 16, 36, 100, 225, 256, 625, 576, 961, 1521, 1444, 2025, 4096, 2304, 2704, 3249, 5625, 6724, 6561, 4900, 5776, 6241, 11236, 12544, 21025, 12321, 14641, 13689, 15129, 17956, 20736, 19321, 21316, 23716, 26569, 36864, 28561, 30976, 32041

Offset: 2

Amarnath Murthy, Oct 20 2002

nonn

David W. Wilson, Table of n, a(n) for n = 2..10000

Cf. A014085, A076957.

a(n) = A076957(n)^2. - David W. Wilson, Jan 08 2017

More terms from Sascha Kurz, Jan 22 2003

Name edited by David W. Wilson, Jan 08 2017

A078763 List primes between (2n-1)^2 and (2n)^2.

Original entry on oeis.org

2, 3, 11, 13, 29, 31, 53, 59, 61, 83, 89, 97, 127, 131, 137, 139, 173, 179, 181, 191, 193, 227, 229, 233, 239, 241, 251, 293, 307, 311, 313, 317, 367, 373, 379, 383, 389, 397, 443, 449, 457, 461, 463, 467, 479, 541, 547, 557, 563, 569, 571, 631, 641, 643, 647

Offset: 1

Donald S. McDonald, Jan 09 2003

Primes are on adjacent sides of the Ulam square prime-spiral.

			The 2 primes between 3^2=9 (odd) and 4^2=16 (even) are just a(3)=11 and a(4)=13.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A014085, A078764.

2,seq(op(select(isprime, [seq(i,i=(2*n-1)^2..(2*n)^2,2)])),n=1..20); # Robert Israel, Oct 28 2020

f[n_] := Select[Range[(2n - 1)^2, (2n)^2], PrimeQ];Flatten@Array[f, 13] (* Ray Chandler, May 03 2007 *)

Extended by Ray Chandler, May 03 2007

A087952 Smallest prime == 1 (mod n) and > n^2.

Original entry on oeis.org

2, 5, 13, 17, 31, 37, 71, 73, 109, 101, 199, 157, 313, 197, 241, 257, 307, 379, 419, 401, 463, 617, 599, 577, 701, 677, 757, 953, 929, 991, 1117, 1153, 1123, 1259, 1471, 1297, 1481, 1483, 1873, 1601, 1723, 1933, 1979, 2069, 2161, 2347, 2351, 2593, 2549, 2551

Offset: 1

Ray Chandler, Sep 16 2003

nonn

Primes arising in A087554.

Since A014085(n) ~ n/log(n) one may conjecture that a(n) < 2*n^2 for all n > 1. Numerically we find a(n) = n^2*(1 + O(1/sqrt(n))). - M. F. Hasler, Feb 27 2020

			For n=1, a(1) = 2, because 2 == 1 mod 1 and 2 > 1^2.
For n=2, a(2) = 5, because 5 == 1 mod 2 and 5 > 2^2.

M. F. Hasler, Table of n, a(n) for n = 1..10000, Feb 27 2020

Cf. A034693, A034694, A087554.

Cf. A014085 (number of primes between n^2 and (n+1)^2).

spr[n_]:=Module[{p=NextPrime[n^2]},While[Mod[p,n]!=1,p=NextPrime[p]];p]; Join[ {2},Array[spr,50,2]] (* Harvey P. Dale, Jun 21 2021 *)

apply( {A087952(n)=forprime(p=n^2+1,,(p-1)%n||return(p))}, [1..66]) \\ M. F. Hasler, Feb 27 2020

Examples added by N. J. A. Sloane, Jun 21 2021

A104477 Number of successive squarefree intervals between primes.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 3, 0, 2, 0, 3, 0, 2, 0, 4, 0, 3, 0, 4, 0, 4, 0, 3, 0, 5, 0, 6, 0, 4, 0, 5, 0, 5, 0, 6, 0, 6, 0, 6, 0, 5, 0, 8, 0, 7, 0, 6, 0, 7, 0, 8, 0, 7, 0, 7, 0, 9, 0, 8, 0, 9, 0, 8, 0, 9, 0, 8, 0, 8, 0, 11, 0, 10, 0, 11, 0, 10, 0, 8, 0, 11, 0, 10, 0, 12, 0, 9, 0, 12, 0, 14, 0, 9, 0, 10, 0

Offset: 1

Alexandre Wajnberg, Apr 18 2005

Find the number (the "run length") of successive intervals [p, p'=nextprime(p)] (followed by [p', p''], then [p'', p'''] etc.) which do not contain a square. When a square (n+1)^2 is found in such an interval, this will result in a term a(2n) = 0, preceded by a(2n-1) = the number of intervals of primes counted before reaching that square, i.e., between n^2 and (n+1)^2. - M. F. Hasler, Oct 01 2018

			a(1)=1 because the first interval between primes (2 to 3) is free of squares.
a(2)=0 because there is a square between 3 and 5.
a(7)=2 because there are two successive squarefree intervals: 17 to 19; and 19 to 23.
a(8)=0 because between 23 and 29 there is a square: 25.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A061265, A031265, A104481.

Equals A014085 - 1 without the initial term, interleaved with 0's.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; PrevPrim[n_] := Block[{k = n - 1}, While[ !PrimeQ[k], k-- ]; k]; f[n_] := If[ EvenQ[n], 0, PrimePi[ PrevPrim[(n + 3)^2/4]] - PrimePi[ NextPrim[(n + 1)^2/4]]]; Table[ f[n], {n, 100}] (* Robert G. Wilson v, Apr 23 2005 *)

p=2; c=0; forprime(np=p+1, 1e4, if( sqrtint(p) < sqrtint(np), print1(c",",c=0,","), c++); p=np) \\ For illustrative purpose. Better:
A104477(n)=if(bittest(n,0),primepi((1+n\/=2)^2)-primepi(n^2)-1,0) \\ M. F. Hasler, Oct 01 2018

a(2n) = 0: this is the interval from the greatest prime less than the (n+1)th square, through that square and up to the least prime greater than that square. - Robert G. Wilson v, Apr 23 2005
a(2n-1) = the difference between the indices of the greatest prime less than (n+1)^2 and the least prime greater than n^2. - Robert G. Wilson v, Apr 23 2005
a(2n-1) = A014085(n) - 1 = primepi((n+1)^2) - primepi(n^2) - 1. - M. F. Hasler, Oct 01 2018

More terms from Robert G. Wilson v, Apr 23 2005

Offset corrected by M. F. Hasler, Oct 01 2018

A132657 a(n) is the product of the least prime > n^2 and the greatest prime < (n+1)^2.

Original entry on oeis.org

6, 35, 143, 391, 899, 1739, 3233, 5293, 8051, 11413, 17653, 24883, 33389, 43931, 56977, 72731, 92881, 118829, 145699, 176039, 212197, 254701, 308911, 357163, 424663, 492179, 566609, 660293, 756611, 864371, 987307, 1120697, 1257923

Offset: 1

Jonathan Vos Post, Nov 15 2007

			a(1) = 6 = 2*3 = (smallest prime in [1^2,2^2]) * (largest prime in [1^2,2^2]).
a(2) = 35 = 5*7 = (smallest prime in [2^2,3^2]) * (largest prime in [2^2,3^2]).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A001358, A007491, A014085, A053000, A053001, A053607, A077766, A077767, A132435.

seq(nextprime(n^2)*prevprime((n+1)^2,n=1..100); # Robert Israel, Jan 26 2020

Table[Prime[PrimePi[n^2] + 1]*Prime[PrimePi[(n + 1)^2]], {n, 1, 40}] (* Stefan Steinerberger, Nov 20 2007 *)
NextPrime[#[[1]]]NextPrime[#[[2]],-1]&/@Partition[Range[40]^2,2,1] (* Harvey P. Dale, Aug 27 2022 *)

for(n=1,33,print1(nextprime(n^2)*precprime((n+1)^2),", ")) \\ Hugo Pfoertner, Jan 26 2020

a(n) = A007491(n) * A053001(n+1).

More terms from Stefan Steinerberger, Nov 20 2007

A144137 Numbers n such that between n^K and (n+1)^K there are no primes, where K = sqrt(2).

Original entry on oeis.org

4, 24, 29, 33, 40, 43, 56, 59, 84, 117, 122, 128, 132, 139, 145, 156, 162, 163, 183, 190, 203, 230, 253, 257, 286, 297, 303, 306, 315, 319, 336, 371, 403, 420, 433, 447, 456, 467, 479, 537, 543, 563, 592, 595, 599, 624, 699, 746, 755, 767, 774, 782, 805, 814

Offset: 1

Artur Jasinski, Sep 11 2008

nonn

			a(1)=4 because in range 4^sqrt(2) = 7.10299... and (4+1)^sqrt(2) = 9.73852... there are no primes (8 and 9 aren't primes).

T. D. Noe, Table of n, a(n) for n=1..133 (no other n < 10^6)

Cf. A014085, A143898, A143935, A144140.

a = {}; k = Sqrt[2]; Do[If[Length[Select[Range[Ceiling[n^k], Floor[(n + 1)^k]], PrimeQ]] == 0, AppendTo[a, n]], {n, 3000}]; a
Select[Range[850],PrimePi[(#+1)^Sqrt[2]]-PrimePi[#^Sqrt[2]]==0&] (* or *) SequencePosition[PrimePi[Range[850]^Sqrt[2]],{x_,x_}][[All,1]] (* Harvey P. Dale, Jul 31 2021 *)

A220506 Number of primes <= n-th quarter-square.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 13, 15, 16, 18, 20, 22, 24, 25, 29, 30, 32, 34, 36, 39, 42, 44, 46, 48, 52, 54, 58, 61, 62, 66, 68, 72, 75, 78, 81, 85, 89, 92, 96, 99, 101, 105, 109, 114, 118, 122, 126, 129, 133, 137, 141, 146, 150, 154, 158, 162, 167, 172, 177, 181, 187, 191, 195, 200

Offset: 1

Omar E. Pol, Feb 05 2013

nonn

Partial sums of A220492. A bisection is A038107, n >= 1.

Cf. A000040, A000720, A002620, A014085, A111208.

Table[PrimePi[n^2/4], {n, 75}] (* Alonso del Arte, Feb 05 2013 *)

a(n) = A000720(A002620(n)), n >= 1.

A222030 Primes and quarter-squares.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 11, 12, 13, 16, 17, 19, 20, 23, 25, 29, 30, 31, 36, 37, 41, 42, 43, 47, 49, 53, 56, 59, 61, 64, 67, 71, 72, 73, 79, 81, 83, 89, 90, 97, 100, 101, 103, 107, 109, 110, 113, 121, 127, 131, 132, 137, 139, 144, 149, 151, 156, 157, 163, 167, 169

Offset: 0

Omar E. Pol, Feb 05 2013

Union of A002620 and A000040.
It appears that there is always a prime between two consecutive quarter squares, if n >= 2. Therefore in a square spiral, or zig-zag, whose vertices are the quarter-squares, it appears that there is always a prime between two consecutive vertices, if n >= 2.
Apparently the above comment is equivalent to the Oppermann's conjecture. - Omar E. Pol, Oct 26 2013
Union of A000040 and A000290 and A002378. - Omar E. Pol, Oct 28 2013

Wikipedia, Oppermann's conjecture

Cf. A000040, A002620, A000290, A014085, A220492, A220506, A220508, A220516.

mx = 13; Union[Prime[Range[PrimePi[mx^2]]], Floor[Range[2*mx]^2/4]] (* Alonso del Arte, Mar 03 2013 *)

a(n) ~ n log n. - Charles R Greathouse IV, Mar 04 2013

A286090 Square matrix m read by antidiagonals. For j <= i, m[i, j] is how often sqrtint(gpf(c)) = j where i^2 <= c < (i+1)^2; for j >= i, m[i, j] is how often sqrtint(p) = i where p is a prime factor of c counted with multiplicity and i^2 <= c < (i+1)^2.

Original entry on oeis.org

3, 7, 3, 8, 2, 2, 15, 3, 3, 3, 15, 2, 2, 2, 2, 21, 6, 1, 1, 4, 2, 19, 3, 2, 3, 2, 3, 1, 27, 8, 2, 1, 1, 2, 5, 2, 29, 7, 2, 2, 2, 2, 2, 3, 2, 31, 7, 4, 2, 0, 0, 2, 4, 3, 1, 32, 9, 3, 3, 2, 4, 2, 3, 3, 4, 1, 41, 9, 3, 3, 0, 0, 0, 0, 3, 3, 4, 2, 39, 11, 6, 4, 2, 1

Offset: 1

David A. Corneth, May 02 2017 and May 17 2017

The greatest prime factor (see A006530) of 1 is 1 by convention. The definitions for m[i, j] where j >= i and j <= i give no contradiction for j = i.

			Let i = 2 and sqrtint(n) = the square root of n rounded down to an integer. The integers c such that i^2 = 2^2 <= c < 3^2 = (i+1)^2 are 4, 5, 6, 7 and 8. The greatest prime factors h of these terms are 2, 5, 3, 7 and 2 respectively. sqrtint(h) are 1, 2, 1, 2 and 1 respectively. 1 occurs thrice here, 2 occurs twice, giving m[2, 1] = 3 and m[2, 2] = 2.
m[1, 2] = 7 because the prime factorizations of 4, 5, 6, 7 and 8 are 2^2, 5, 2*3, 7, 2^3. The prime 2 occurs 6 times and sqrtint(2) = 1. The prime factor 3 occurs once and sqrtint(3) = 1. Therefore m[1, 2] = 6 + 1 = 7. p = 5 and p = 7 each occur once and each have sqrtint(p) = 2. Therefore m[2, 2] = 2 as found earlier.
The first block of m of 10 by 10 is:
[3 7 8 15 15 21 19 27 29 31]
[3 2 3  2  6  3  8  7  7  9]
[2 3 2  1  2  2  2  4  3  3]
[3 2 1  3  1  2  2  3  3  4]
[2 4 2  1  2  0  2  0  2  1]
[2 3 2  2  0  4  0  1  3  1]
[1 5 2  2  2  0  3  0  0  2]
[2 3 4  3  0  1  0  4  0  0]
[2 3 3  3  2  3  0  0  3  0]
[1 4 3  4  1  1  2  0  0  5]

David A. Corneth, Table of n, a(n) for n = 1..10011 (first 142 rows flattened.)

Cf. A006530 (greatest prime factor, gpf), A014085.

squaremat(n) = {my(m=matrix(n,n)); m[1,1] = 3; for(i=2, n, for(c = i^2, (i+1)^2-1, f=factor(c); for(j=1, matsize(f)[1], m[sqrtint(f[j,1]), i] += f[j, 2]); m[i, sqrtint(f[matsize(f)[1], 1])]++));for(i=2, n, m[i,i] = m[i,i] >> 1); m}
\\ Diagonals m[i, j] where i + j <= n.
upto(n) = {my(v=vector(binomial(n+1, 2)), m = squaremat(n), t = 0); for(i=1, n, for(j=0, i-1, t++; v[t] = m[j+1, i-j])); v}

for n > 1, m[n, n] = A014085(n).

A320688 Sum of the square excess A056892 of the primes between two squares.

Original entry on oeis.org

3, 4, 6, 11, 10, 24, 26, 34, 26, 33, 50, 67, 72, 46, 70, 109, 96, 132, 122, 153, 132, 145, 174, 229, 208, 175, 194, 287, 232, 244, 338, 267, 276, 345, 374, 239, 392, 396, 424, 390, 484, 373, 514, 563, 618, 424, 654, 821, 442, 557, 890, 814, 668, 741, 580, 642, 990, 811, 982, 968, 772

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Consider the primes p1,...,pK between two squares n^2 and (n+1)^2, and take the sum of the differences: (p1 - n^2) + ... + (pK - n^2). Obviously this equals (sum of these primes) - (number of these primes) * n^2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A014085, A000290, A108314, A106044.

Row sums of A056892, read as a table.

R:= NULL: p:= 2: n:= 1: t:= 0:
while n <= 100 do
    t:= t + p-n^2;
    p:= nextprime(p);
    if p > (n+1)^2 then
     R:= R, t; t:= 0; n:= n+1;
    fi:
od:
R; # Robert Israel, Dec 17 2024

a(n,s=0)={forprime(p=n^2,(n+1)^2,s+=p-n^2);s}

a(n) = A108314(n) - A014085(n)*A000290(n), where A000290(n) = n^2.

cp's OEIS Frontend

A076956 Smallest k^2 such that there are exactly n primes between k^2 and (k+1)^2.

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A078763 List primes between (2n-1)^2 and (2n)^2.

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Maple

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A087952 Smallest prime == 1 (mod n) and > n^2.

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PARI

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A104477 Number of successive squarefree intervals between primes.

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A132657 a(n) is the product of the least prime > n^2 and the greatest prime < (n+1)^2.

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A144137 Numbers n such that between n^K and (n+1)^K there are no primes, where K = sqrt(2).

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Mathematica

A220506 Number of primes <= n-th quarter-square.

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Mathematica

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A222030 Primes and quarter-squares.

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