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Leading diagonal of triangle A119446.
3

%I #6 Apr 07 2023 17:16:36

%S 2,2,3,3,3,3,3,3,3,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,13,13,7,7,7,7,13,13,

%T 13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,

%U 13,13,13,13,13,13,13,13,13,13,13,19,19,19,19,19,19,19,19,19,19,19,19,19

%N Leading diagonal of triangle A119446.

%C a(181) = 27 is the first term greater than 19. This is because prime(181)/181 > 6 for the first time. In general this sequence is determined by prime(n)/n: the pattern for each row of the triangle is that it ends with prime(n), preceded by multiples of k = prime(n)/n down to k^2, then the largest multiple of k-1 less than k^2 and the largest multiple of k-2 less than that and so on. This sequence gives the multiple of 1. See A000960 for the sequence that gives the ending value for each starting k.

%H G. C. Greubel, <a href="/A119447/b119447.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = A000960( prime(n)/n ).

%F a(n) = A119446(n, n).

%t t[1, n_]:= Prime[n];

%t t[m_, n_]/; 1<m<=n:= t[m,n]= (n-m+1)*Floor[(t[m-1,n]-1)/(n-m+1)];

%t t[_, _]=0;

%t A119447[n_]:= A119447[n]= t[n,n];

%t Table[A119447[n], {n,100}] (* _G. C. Greubel_, Apr 07 2023 *)

%o (Magma)

%o function t(n,k) // t = A119446

%o if k eq 1 then return NthPrime(n);

%o else return (n-k+1)*Floor((t(n,k-1) -1)/(n-k+1));

%o end if;

%o end function;

%o [t(n,n): n in [1..100]]; // _G. C. Greubel_, Apr 07 2023

%o (SageMath)

%o def t(n, k): # t = A119446

%o if (k==1): return nth_prime(n)

%o else: return (n-k+1)*((t(n, k-1) -1)//(n-k+1))

%o def A119447(n): return t(n,n)

%o [A119447(n) for n in range(1,101)] # _G. C. Greubel_, Apr 07 2023

%Y Cf. A000960, A119444, A119446.

%K nonn

%O 1,1

%A _Joshua Zucker_, May 20 2006