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

1,1

Comments

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

References

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

Links

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

Example

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.

Mathematica

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 *)

Prog

(Magma)
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]];

Crossrefs

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

Cf. A277644 and A277645.

Sequence in context: A285047 A263137 A320846 * A247202 A195758 A304684

Adjacent sequences: A277512 A277513 A277514 * A277516 A277517 A277518

Keyword

nonn,easy

Author

Jason Kimberley, Oct 18 2016

Status

approved