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A278113 Triangle T(n,k) = A278112(n,A000040(k)) for 1 <= k <= A278114(n), read by rows.

%I A278113 #24 Nov 19 2024 22:11:54
%S A278113 1,2,1,1,1,3,2,1,1,1,1,1,4,3,2,2,1,1,1,1,1,1,1,5,4,3,2,2,1,1,1,1,1,1,
%T A278113 1,1,1,1,6,4,3,3,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,7,5,4,3,2,2,2,2,2,1,
%U A278113 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,8,6,5,4,3,3,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1
%N A278113 Triangle T(n,k) = A278112(n,A000040(k)) for 1 <= k <= A278114(n), read by rows.
%C A278113 This triangle consists of those columns of A278112 that have prime index.
%H A278113 Jason Kimberley, <a href="/A278113/b278113.txt">Table of n, a(n) for n = 1..10126</a> (rows 1..46)
%F A278113 T(n,k) = floor(n*sqrt(2/prime(k))).
%F A278113 T(n,k) sqrt(A000040(k)) <= n sqrt(2) < (T(n,k)+1) sqrt(A000040(k)).
%e A278113 The first eight rows are:
%e A278113   1;
%e A278113   2, 1, 1, 1;
%e A278113   3, 2, 1, 1, 1, 1, 1;
%e A278113   4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1;
%e A278113   5, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%e A278113   6, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%e A278113   7, 5, 4, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%e A278113   8, 6, 5, 4, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%t A278113 Table[Floor[n Sqrt[2/Prime@ k]], {n, 8}, {k, PrimePi[2 n^2]}] // Flatten (* _Michael De Vlieger_, Feb 17 2017 *)
%o A278113 (Magma)
%o A278113 A278112:=func<n,k|Isqrt(2*n^2 div k)>;
%o A278113 A278113_row:=func<n|[A278112(n,p):p in PrimesUpTo(2*n^2)]>;
%o A278113 &cat[A278113_row(n):n in[1..8]];
%Y A278113 Cf. A000040, A277648, A278112, A278114, A278115, A278118.
%K A278113 nonn,tabf,easy
%O A278113 1,2
%A A278113 _Jason Kimberley_, Feb 09 2017