A119447 Leading diagonal of triangle A119446.

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

Offset: 1

Author

Joshua Zucker, May 20 2006

Keywords

• nonn

Comments

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.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000960, A119444, A119446.

Programs

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

Mathematica

t[1, n_]:= Prime[n];
t[m_, n_]/; 1, ]=0;
A119447[n_]:= A119447[n]= t[n,n];
Table[A119447[n], {n,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

Formula

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

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