A084597 -id:A084597 - cp's OEIS frontend

A349999 Least number m of primes that must have appeared in an interval [j^2, (j+1)^2], such that all intervals [k^2, (k+1)^2], k>j contain more than m primes. The corresponding values of j are A349998.

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 18, 19, 22, 24, 26, 27, 28, 29, 30, 32, 33, 35, 36, 38, 39, 40, 41, 44, 45, 47, 51, 54, 56, 63, 65, 68, 70, 71, 78, 80, 85, 94, 99, 106, 107, 114, 115, 120, 121, 127, 133, 138, 146, 154, 155, 164, 168, 169, 175, 176, 177

Offset: 1

Hugo Pfoertner, Dec 09 2021

nonn

All terms are empirical (see the graph of A014085 for the limited width of the scatter band), but supporting the validity of Legendre's conjecture that there is always a prime between n^2 and (n+1)^2.

The terms are determined by searching from large to small indices in A014085 for new minima.

			See A349997 and A349998.

Hugo Pfoertner, Table of n, a(n) for n = 1..2414

Cf. A014085, A084597, A349997, A349998.

Cf. A333846, A349996.

a(n) = A014085(A349998(n)).
A014085(k) > a(n) for k > A349998(n).
A014085(k) >= a(n) for k >= A349997(n).

A076957 Smallest k such that there are exactly n primes strictly between k^2 and (k+1)^2.

Original entry on oeis.org

1, 4, 6, 10, 15, 16, 25, 24, 31, 39, 38, 45, 64, 48, 52, 57, 75, 82, 81, 70, 76, 79, 106, 112, 145, 111, 121, 117, 123, 134, 144, 139, 146, 154, 163, 192, 169, 176, 179, 193, 202, 218, 204, 226, 223, 240, 233, 238, 243, 259, 291, 256, 286, 309, 278

Offset: 2

Amarnath Murthy, Oct 20 2002

nonn

From David W. Wilson, Jan 08 2017: (Start)
a(n)^2 = A076956(n).
A014085(a(n)) = n.
Conjecturally, a(n) is undefined for n = 1 and defined for all n >= 2. (End)

T. D. Noe and David W. Wilson, Table of n, a(n) for n = 2..10000 (a(n) for n = 2..1000 from T. D. Noe).

Cf. A014085, A076956, A084597.

a := proc(n) local k, h, SEARCHLIMIT; SEARCHLIMIT := 10000; h := proc(k) option remember; nops(select(j->isprime(j), [$k^2+1..(k+1)^2])) end: k := 1; while h(k) <> n and k < SEARCHLIMIT do k := k+1 od; `if`(k=SEARCHLIMIT, print("Search limit reached!"), k) end: seq(a(n), n=2..56); # Peter Luschny, Jan 10 2017

Table[k = 1; While[Count[Map[PrimeQ, Range[k^2 + 1, (k + 1)^2]], True] != n, k++]; k, {n, 2, 56}] (* Michael De Vlieger, Jan 10 2017 *)
With[{pp=Table[Count[Range[n^2+1,(n+1)^2-1],?(PrimeQ[#]&)],{n,500}]},Table[ Position[pp,k,1,1],{k,60}]]//Flatten (* _Harvey P. Dale, Aug 01 2021 *)

More terms from Ralf Stephan, Oct 31 2002

A333846 Numbers k such that the number of primes between k^2 and (k+1)^2 increases to a new record.

Original entry on oeis.org

0, 1, 4, 6, 10, 15, 16, 24, 31, 38, 45, 48, 52, 57, 70, 76, 79, 106, 111, 117, 123, 134, 139, 146, 154, 163, 169, 176, 179, 193, 202, 204, 223, 233, 238, 243, 256, 278, 284, 318, 326, 336, 359, 369, 412, 419, 430, 456, 458, 468, 479, 517, 550, 564, 595, 601, 612

Offset: 1

Bernard Schott, Apr 08 2020

nonn

Legendre's conjecture (still open) states that for n > 0 there is always a prime between n^2 and (n+1)^2. The number of primes between n^2 and (n+1)^2 is equal to A014085(n), so, the corresponding records are given by A014085(a(n)) = 0, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, ... (A349996).
m = 25 is the smallest number such that there are exactly 8 primes between m^2 = 625 et (m+1)^2 = 676, namely {631, 641, 643, 647, 653, 659, 661, 673} but there are 9 primes between 24^2 = 576 et 25^2 = 625, namely {577, 587, 593, 599, 601, 607, 613, 617, 619} so 24 is a term but not 25; hence, 25 is the first term of A076957 that is not a record.
This sequence is infinite. Suppose for contradiction that a(n) = k was the last term, with s primes between k^2 and (k+1)^2. Then there are at most s primes between (k+1)^2 and (k+2)^2, at most s primes between (k+2)^2 and (k+3)^3, and at most s*sqrt(x) + pi(k^2) primes up to x. But there are ~ x/log x primes up to x by the Prime Number Theorem, a contradiction. This can be made sharp with various explicit estimates. - Charles R Greathouse IV, Apr 10 2020

			There are 7 primes between 16^2 and 17^2, i.e., 256 and 289, which are 257, 263, 269, 271, 277, 281, 283, and there does not exist k < 16 with 7 or more primes between k^2 and (k+1)^2, hence, 16 is in the sequence.

Hugo Pfoertner, Table of n, a(n) for n = 1..2533
Mac Tutor History of Mathematics, Adrien-Marie Legendre
Wikipedia, Legendre's conjecture

Cf. A014085, A076957, A084597.
Cf. A333241 (Similar records between k and (9/8)*k).
Cf. A349996, A349997, A349998, A349999.

primeCount[n_] := PrimePi[(n + 1)^2] - PrimePi[n^2]; pmax = -1; seq = {}; Do[p = primeCount[n]; If[p > pmax, pmax = p; AppendTo[seq, n]], {n, 0, 612}]; seq (* Amiram Eldar, Apr 08 2020 *)

print1(pr=0,", ");pp=0;for(k=1,650,my(pc=primepi(k*k));if(pc-pp>pr,print1(k-1,", ");pr=pc-pp);pp=pc) \\ Hugo Pfoertner, Apr 10 2020

More terms from Michel Marcus, Apr 08 2020

A084596 a(n) is the number of times n is in sequence A014085; i.e., there are exactly a(n) cases where there are exactly n primes between m^2 and (m+1)^2 for m >= 0.

Original entry on oeis.org

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

Offset: 0

Harry J. Smith, May 31 2003

nonn

This sequences uses a finite number of terms of A014085 to conjecture the behavior of all terms of A014085. The first 10000 terms of this sequence were computed using 120000 terms of A014085. Legendre's conjecture is equivalent to a(0) remaining 1 for all terms of A014085. [Comment reworded by T. D. Noe, Sep 04 2008]

			a(14)=2 because 14 is in sequence A014085 only two times. There are 14 primes between 64^2 and 65^2 as well as between 77^2 and 78^2. These are the only cases with exactly 14 primes.

P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 143.

T. D. Noe, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Landau's Problems.

Cf. A007491, A014085, A084597.

cp's OEIS Frontend

A349999 Least number m of primes that must have appeared in an interval [j^2, (j+1)^2], such that all intervals [k^2, (k+1)^2], k>j contain more than m primes. The corresponding values of j are A349998.

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A076957 Smallest k such that there are exactly n primes strictly between k^2 and (k+1)^2.

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A333846 Numbers k such that the number of primes between k^2 and (k+1)^2 increases to a new record.

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A084596 a(n) is the number of times n is in sequence A014085; i.e., there are exactly a(n) cases where there are exactly n primes between m^2 and (m+1)^2 for m >= 0.

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