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

%I A277515 #17 Dec 31 2016 01:46:58
%S A277515 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,
%T A277515 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,
%U A277515 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
%N A277515 Smallest prime p such that n sqrt(2) < m sqrt(p) < (n+1) sqrt(2) for some integer m.
%C A277515 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.
%C A277515 Eggleton et al. show that a(n)=3 if and only if n is a term in A277644.
%C A277515 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
%D A277515 R. B. Eggleton, J. S. Kimberley and J. A. MacDougall, Square-free rank of integers, submitted.
%H A277515 Jason Kimberley, <a href="/A277515/b277515.txt">Table of n, a(n) for n = 1..10000</a>
%e A277515 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.
%t A277515 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 *)
%o A277515 (Magma)
%o A277515 function A277515(n)
%o A277515   p := 2;
%o A277515   lower := 2*n^2;
%o A277515   upper := 2*(n+1)^2;
%o A277515   repeat
%o A277515     p := NextPrime(p);
%o A277515     m := Isqrt(upper div p);
%o A277515   until p*m^2 gt lower;
%o A277515   return p;
%o A277515 end function;
%o A277515 [A277515(n):n in[1..100]];
%Y A277515 First occurrences of each prime > 2 are listed in A278107.
%Y A277515 Cf. A277644 and A277645.
%K A277515 nonn,easy
%O A277515 1,1
%A A277515 _Jason Kimberley_, Oct 18 2016