A277515 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

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;