A053001 -id:A053001 - cp's OEIS frontend

A331698 a(n) = (1/2) * ((greatest prime < (n+1)^2) - (least prime > n^2)) for n >= 2.

1, 1, 3, 1, 5, 4, 6, 7, 6, 6, 9, 10, 13, 12, 13, 12, 14, 15, 19, 18, 18, 15, 21, 21, 25, 20, 26, 17, 23, 27, 28, 31, 30, 31, 35, 33, 32, 37, 34, 33, 35, 36, 34, 43, 39, 42, 45, 33, 45, 45, 48, 45, 53, 42, 46, 55, 49, 51, 56, 53, 60, 52, 60, 60, 63, 64, 61, 53

Offset: 2

Hugo Pfoertner, Jan 27 2020

nonn

			a(2) = 1 because 7 is the greatest prime < 3^2 and 5 is the least prime > 2^2. (7-5)/2 = 1.

Hugo Pfoertner, Table of n, a(n) for n = 2..10000

Cf. A007491, A053001, A132657.

for(n=2, 69, print1((precprime((n+1)^2)-nextprime(n^2))/2, ", "))

A365706 For n >= 1, a(n) is the least prime p such that the arithmetic mean of (n + 1) consecutive primes starting with p is a perfect square, or a(n) = -1 if no such p exists.

Original entry on oeis.org

3, 2393, 5, 827, 53, 271, 1063, 23993, 197, 29, 193, 2143, 359, 6829, 397, 17, 433, 661, 2837, 25171, 13597, 563, 10301, 1814233, 51427, 6781, 316817, 7477, 71, 238919, 11491, 3109, 42293, 38653, 6263, 13043, 474497, 21433, 13, 21419, 16963, 5119, 705209, 183761

Offset: 1

Ctibor O. Zizka, Oct 15 2023

nonn

Does a(n) exists for all n >= 1 ?
From David A. Corneth, Oct 18 2023: (Start)
Let s(n) be the sum of n + 1 consecutive primes starting with a(n). Then s(n)/(n+1) = m^2 for some positive integer m.
This means s(n) = (n + 1) * m^2. If n is even then m is odd if a(n) > 2.
As s(n) >= A007504(n) we have m^2 >= s(n)/(n+1) >= A007504(n)/(n+1) i.e. m >= sqrt(A007504(n)/(n+1)). So for some m we can see if m^2 * (n+1) is the sum of n+1 consecutive primes and if so a(n) is the smallest prime of these n+1 primes after testing all candidates up to m. (End)
s(n) = (n + 1)* a(n) + Sum_{i=0..(n-1)} (n-i)*g(i+1), thus we have Sum_{i=0..(n-1)} (n-i)*g(i+1) = (m^2 - a(n)) * (n + 1), g(j) are the n gaps between n + 1 consecutive primes. - Ctibor O. Zizka, Oct 18 2023

			n = 2: we search for the least prime(i) such that (prime(i) + prime(i + 1) + prime(i + 2))/3 = m^2, m an integer. This is valid for (2393 + 2399 + 2411)/3 = 49^2 thus a(2) = 2393.

David A. Corneth, Table of n, a(n) for n = 1..1000

Cf. A000040, A007504, A053001, A225195.

isok(x) = (denominator(x)==1) && issquare(x);
a(n) = my(k=1); while (!isok((vecsum(primes(k+n))-vecsum(primes(k-1)))/(n+1)), k++); prime(k); \\ Michel Marcus, Oct 16 2023

a(n) = {my(m = n + 1, ps = vector(m, i, prime(i)), s); forprime(p = ps[m] + 1, , s = vecsum(ps); if(!(s % m) && issquare(s/m), return(ps[1])); ps = concat(vecextract(ps, "^1"), p));} \\ Amiram Eldar, Oct 18 2023

More terms from Amiram Eldar, Oct 18 2023

A379444 a(n) is the difference between the least prime > (n+1)^2 and the largest prime < n^2, divided by 2.

Original entry on oeis.org

4, 5, 8, 7, 11, 10, 11, 11, 15, 18, 17, 15, 17, 17, 21, 24, 25, 21, 23, 24, 31, 27, 30, 29, 30, 30, 40, 34, 40, 39, 35, 38, 38, 37, 41, 40, 42, 45, 48, 54, 51, 51, 47, 56, 50, 51, 57, 52, 66, 57, 60, 57, 64, 57, 65, 71, 65, 69, 67, 64, 78, 66, 68, 69, 72, 77, 81

Offset: 2

Hugo Pfoertner, Dec 23 2024

2*a(n) would be the gap needed between consecutive primes to provide a counterexample to Legendre's conjecture that there is always a prime between n^2 and (n+1)^2. The gaps actually observed are significantly smaller; see A378904 for comparison.

Hugo Pfoertner, Table of n, a(n) for n = 2..10000

Cf. A001223, A007491, A053001, A058043, A350100, A378904.

a[n_]:=(NextPrime[(n+1)^2] - NextPrime[n^2,-1])/2; Array[a,67,2] (* Stefano Spezia, Jan 24 2025 *)

a379444(n) = (nextprime((n+1)^2) - precprime(n^2))/2

a(n) = (A007491(n+1) - A053001(n))/2.

a(n) >= n + 2.

A173832 Largest prime < n^5.

Original entry on oeis.org

31, 241, 1021, 3121, 7759, 16787, 32749, 59029, 99991, 161047, 248827, 371291, 537811, 759371, 1048573, 1419839, 1889561, 2476081, 3199997, 4084081, 5153623, 6436327, 7962607, 9765619, 11881357, 14348891, 17210353, 20511143, 24299981

Offset: 2

Vladimir Joseph Stephan Orlovsky, Feb 25 2010

nonn

Cf. A014220, A053001, A077037, A173831

PrimePrev[n_]:=Module[{k},k=n-1;While[ !PrimeQ[k],k-- ];k];f[n_]:=n^5;lst={};Do[AppendTo[lst,PrimePrev[f[n]]],{n,5!}];lst
NextPrime[Range[2, 30]^5, -1]  (* Ray Chandler, Dec 08 2018 *)

A253474 Numbers n such that the difference between n^2 and largest prime less than n^2 is not prime.

Original entry on oeis.org

2, 11, 17, 23, 25, 31, 39, 41, 45, 51, 53, 56, 57, 59, 65, 67, 73, 76, 79, 81, 83, 85, 87, 91, 95, 97, 99, 100, 101, 105, 109, 111, 113, 115, 123, 125, 129, 133, 137, 141, 143, 147, 149, 151, 153, 154, 157, 159, 163, 165, 167, 170, 171, 175, 179, 181, 185, 187, 189, 193, 195, 197, 199, 201, 203, 207, 209, 213, 215, 219, 221, 225

Offset: 1

Carlos Eduardo Olivieri, Mar 17 2015

Indices of terms in A056927 that are not prime.

			a(1) = 2, since 2^2 - 3 = 1.
a(2) = 11, since 11^2 - 113 = 8.
a(3) = 17, since 17^2 - 283 = 6.
a(4) = 23, since 23^2 - 523 = 6.

Richard L. Francis, Between consecutive squares, Missouri Journal of Mathematical Sciences, Volume 16 Issue #1 - Winter 2004.

Cf. A000040, A000290, A056927, A053001.

f[n_] := n^2 - NextPrime[n^2, -1]; Select[Range[2, 230], !PrimeQ[f[#]] &]

lista(nn) = for (n=2, nn, if (!isprime(n^2-precprime(n^2)), print1(n, ", "))); \\ Michel Marcus, Mar 22 2015

cp's OEIS Frontend

A331698 a(n) = (1/2) * ((greatest prime < (n+1)^2) - (least prime > n^2)) for n >= 2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A365706 For n >= 1, a(n) is the least prime p such that the arithmetic mean of (n + 1) consecutive primes starting with p is a perfect square, or a(n) = -1 if no such p exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Extensions

A379444 a(n) is the difference between the least prime > (n+1)^2 and the largest prime < n^2, divided by 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A173832 Largest prime < n^5.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A253474 Numbers n such that the difference between n^2 and largest prime less than n^2 is not prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI