A278116 - OEIS

%I #14 Sep 08 2022 08:46:17

%S 1,2,3,3,4,3,2,2,5,4,4,2,2,3,3,5,3,2,2,4,3,3,2,2,3,4,6,6,2,3,4,3,3,2,

%T 2,3,5,4,4,2,4,3,4,3,2,2,3,4,3,2,2,4,3,4,3,2,2,3,4,3,2,2,3,3,5,3,2,2,

%U 4,5,4,2,2,3,3,4,3,2,3,4,7,5,2,2,3,4,2,2,2,3,5,5,5,2,2,3,4,3,2,2,4,5,3,3,2

%N a(n) is the largest j such that A278115(n,k) strictly decreases for k=1..j.

%H Jason Kimberley, <a href="/A278116/b278116.txt">Table of n, a(n) for n = 1..100000</a>

%t Map[1 + Length@ TakeWhile[Differences@ #, # < 0 &] &, #] &@ Table[# Floor[n Sqrt[2/#]]^2 &@ Prime@ k, {n, 105}, {k, PrimePi[2 n^2]}] (* _Michael De Vlieger_, Feb 17 2017 *)

%o (Magma)

%o A:=func<n,k|Isqrt(2*n^2 div k)^2*k>;

%o A278116:=func<n|(exists(j){j:j in[1..#P-1]|A(n,P[j])le A278115(n,P[j+1])}

%o   select j else #P) where P is PrimesUpTo(2*n^2)>;

%o [A278116(n):n in[1..103]];

%o (Python)

%o def isqrt(n):

%o     if n < 0:

%o         raise ValueError('imaginary')

%o     if n == 0:

%o         return 0

%o     a, b = divmod(n.bit_length(),2)

%o     x = 2**(a+b)

%o     while True:

%o         y = (x + n//x)//2

%o         if y >= x:

%o             return x

%o         x = y;

%o def next_prime(n):

%o     for p in range(n+1,2*n+1):

%o         for i in range(2,isqrt(n)+1):

%o             if p % i == 0:

%o                 break

%o         else:

%o             return p

%o     return None

%o def A278116(n):

%o     k = 0

%o     p = 2

%o     s2= (n**2)*p

%o     s = s2

%o     while True:

%o         s_= s

%o         k+= 1

%o         p = next_prime(p)

%o         s = (isqrt(s2//p)**2)*p

%o         if s > s_:

%o             break

%o     return k

%Y Cf. A278102.

%Y This is the row length sequence for triangles A278117 and A278118.

%Y A278119 lists first occurrences in this sequence.

%K nonn,easy

%O 1,2

%A _Jason Kimberley_, Feb 12 2017