A278107 -id:A278107 - cp's OEIS frontend

A277515 Smallest prime p such that n sqrt(2) < m sqrt(p) < (n+1) sqrt(2) for some integer m.

3, 3, 3, 3, 7, 3, 3, 3, 3, 13, 3, 3, 3, 3, 3, 7, 3, 3, 3, 3, 11, 3, 3, 3, 3, 3, 19, 3, 3, 3, 3, 11, 3, 3, 3, 3, 3, 13, 3, 3, 3, 3, 7, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 7, 3, 3, 3, 3, 7, 3, 3, 3, 3, 3, 7, 3, 3, 3, 3, 11, 3, 3, 3, 3, 3, 7, 3, 3, 3, 3, 13, 3, 3, 3, 3, 3, 7, 3, 3, 3, 3, 19, 3, 3, 3, 3, 3, 5, 3, 3

Offset: 1

Jason Kimberley, Oct 18 2016

In fact, m is both the ceiling of the square root of 2n^2/p and the floor of the square root of 2(n+1)^2 / p.
Eggleton et al. show that a(n)=3 if and only if n is a term in A277644.
First occurrence of the n-th prime > 2: 1, 49, 5, 21, 10, 174, 27, 223, 1656, 3901, 1286, 1847, 5095, 3117, 5678, 1727, 14844, 23678, 10986, 33868, 41241, 42794, 50451, 35301, 39546, 206241, 10561, 89600, 50075, 87273, 75922, 142760, 3493, 236213, 277242, 805287, 619149, 333339, 308517, 186105, 109981, 1385669, 215516, 1389450, 130253, 29797, 368004, 584234, 879460, 1711711, 6061772, 2401437, 1891953, 3664144, 1465847, 3260206, 2908877, 4414026, 1338945, 506017, 5420710, ..., . - Robert G. Wilson v, Nov 17 2016

			a(5)=7 because 3 r(5) < 4 r(3) < 5 r(2) < 3 r(7) < 6 r(2) < 5 r(3) < 4 r(5), where r(x) is the square root of x.

R. B. Eggleton, J. S. Kimberley and J. A. MacDougall, Square-free rank of integers, submitted.

Jason Kimberley, Table of n, a(n) for n = 1..10000

First occurrences of each prime > 2 are listed in A278107.

Cf. A277644 and A277645.

function A277515(n)
  p := 2;
  lower := 2*n^2;
  upper := 2*(n+1)^2;
  repeat
    p := NextPrime(p);
    m := Isqrt(upper div p);
  until p*m^2 gt lower;
  return p;
end function;
[A277515(n):n in[1..100]];

f[n_] := Block[{p = 2}, While[ Ceiling[ Sqrt[2 n^2/p]] != Floor[ Sqrt[2 (n + 1)^2/p]], p = NextPrime@ p]; p]; Array[f, 80] (* Robert G. Wilson v, Nov 17 2016 *)

cp's OEIS Frontend

A277515 Smallest prime p such that n sqrt(2) < m sqrt(p) < (n+1) sqrt(2) for some integer m.

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