proposed

approved

editing

proposed

Leading diagonal of triangle A119446, as described in A100461, except with a(1,n) = prime(n) instead of 2^(n-1).

Leading diagonal of triangle A119446.

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

With k = prime(n)/n, a(n) = A000960(k).

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

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

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

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

t[_, _]=0;

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

Table[A119447[n], {n, 100}] (* G. C. Greubel, Apr 07 2023 *)

(Magma)

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

if k eq 1 then return NthPrime(n);

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

end if;

end function;

[t(n, n): n in [1..100]]; // G. C. Greubel, Apr 07 2023

(SageMath)

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

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

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

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

[A119447(n) for n in range(1, 101)] # G. C. Greubel, Apr 07 2023

Cf. A100461 for powers of 2, A119444 for Fibonacci and A119446 for triangle corresponding to this diagonal.

Cf. A000960, A119444, A119446.

approved

editing

_Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), _, May 20 2006

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.

Cf. A100461 for powers of 2, A119444 for Fibonacci, and A119446 for triangle corresponding to this diagonal.

nonn,new

nonn

Leading diagonal of triangle A119446, as described in A100461, except with a(1,n) = prime(n) instead of 2^(n-1).

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, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19

1,1

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.

With k = prime(n)/n, a(n) = A000960(k).

Cf. A100461 for powers of 2, A119444 for Fibonacci, and A119446 for triangle corresponding to this diagonal.

nonn

Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), May 20 2006

approved