cp's OEIS Frontend

A278118 Irregular triangle T(n,k) = A278113(n,k) for 1 <= k <= A278116(n), read by rows.

%I A278118 #10 Sep 08 2022 08:46:17
%S A278118 1,2,1,3,2,1,4,3,2,5,4,3,2,6,4,3,7,5,8,6,9,7,5,4,3,10,8,6,5,11,8,6,5,
%T A278118 12,9,13,10,14,11,8,15,12,9,16,13,10,8,6,17,13,10,18,14,19,15,20,16,
%U A278118 12,10,21,17,13,22,17,13,23,18,24,19,25,20,15,26,21,16,13,27,22,17,14,11,10
%N A278118 Irregular triangle T(n,k) = A278113(n,k) for 1 <= k <= A278116(n), read by rows.
%C A278118 This triangle lists the "descending sequences for rank 1" of Eggleton et al.
%D A278118 R. B. Eggleton, J. S. Kimberley and J. A. MacDougall, Square-free rank of integers, submitted.
%H A278118 Jason Kimberley, <a href="/A278118/b278118.txt">Table of i, a(i) for i = 1..10104 (T(n,k) for n = 1..3333)</a>
%F A278118 From A278113: T(n,k) sqrt(prime(k)) <= n sqrt(2) < (T(n,k)+1) sqrt(prime(k)).
%F A278118 Here, we also have:
%F A278118 T(n,1) sqrt(2) > T(n,2) sqrt(3) > ... > T(n,A278116(n)) sqrt(prime(A278116(n))).
%e A278118 For example, 6 sqrt(2) > 4 sqrt(3) > 3 sqrt(5), because 72 > 48 > 45.
%e A278118 The first six rows are:
%e A278118 1;
%e A278118 2, 1;
%e A278118 3, 2, 1;
%e A278118 4, 3, 2;
%e A278118 5, 4, 3, 2;
%e A278118 6, 4, 3;
%t A278118 Function[w, MapIndexed[Take[w[[First@ #2, 1]], 1 + Length@ TakeWhile[ Differences@ #1, # < 0 &]] &, w[[All, -1]]]]@ Table[Function[k, Function[p, {#, p #^2} &@ Floor[n Sqrt[2/p]]]@ Prime@ k]@ Range@ PrimePi[2 n^2], {n, 27}] (* _Michael De Vlieger_, Feb 17 2017 *)
%o A278118 (Magma)
%o A278118 A278112:=func<n,k|Isqrt(2*n^2 div k)>;
%o A278118 A278115_row:=func<n|[A278112(n,p)^2*p:p in PrimesUpTo(2*n^2)]>;
%o A278118 A278116:=func<n|(exists(j){j:j in[1..#row-1]|row[j]le row[j+1]}select j else #row) where row is A278115_row(n)>;
%o A278118 A278118_row:=func<n|[A278112(n,NthPrime(k)):k in[1..A278116(n)]]>;
%o A278118 [A278118_row(n):n in[1..20]];
%Y A278118 Cf. A278104.
%K A278118 nonn,tabf,easy
%O A278118 1,2
%A A278118 _Jason Kimberley_, Feb 12 2017