cp's OEIS Frontend

A353010 a(n) = maximal d such that Product_{k=0..m} binomial(m,k) is divisible by m^(m+d), where m = A276710(n).

%I A353010 #24 Jun 09 2022 11:33:46
%S A353010 0,0,3,0,1,9,49,0,21,19,31,73,0,61,57,16,4,46,13,43,25,0,20,106,1,57,
%T A353010 172,81,43,66,25,29,51,41,38,140,80,1,71,0,0,34,117,59,199,134,208,
%U A353010 181,9,55,259,202,114,28,263,100,145,32,157,217,60,121,36,73,86,94,19,67,154,21,40,73,57,167,392,135,256
%N A353010 a(n) = maximal d such that Product_{k=0..m} binomial(m,k) is divisible by m^(m+d), where m = A276710(n).
%C A353010 By definition of A276710, a(n) >= -1.
%C A353010 It is conjectured that a(n) >= 0, computationally verified up to n = 10^7.
%C A353010 Empirically from terms up to n=10^7, a(n) seems to become quite large, small values are rare, and yet a(n)=0 also seems to occur for large n.
%e A353010 The 7th term of A276710 is 105 because Product_{k=1..105} binomial(36,k) is divisible by 105^(105-1). Actually, it is divisible by 105^(105+49), but not by 105^(105+50). Therefore, a(7) = 49.
%o A353010 (Python)
%o A353010 from math import prod, comb
%o A353010 from itertools import islice
%o A353010 from sympy import nextprime
%o A353010 def A353010_gen(): # generator of terms
%o A353010     p, q = 3, 5
%o A353010     while True:
%o A353010         for m in range(p+1,q):
%o A353010             r = m**(m-1)
%o A353010             c = 1
%o A353010             for k in range(m+1):
%o A353010                 c = c*comb(m,k) % r
%o A353010             if c == 0:
%o A353010                 d, (e, f) = -m, divmod(prod(comb(m,k) for k in range(m+1)),m)
%o A353010                 while f == 0:
%o A353010                     d += 1
%o A353010                     e, f = divmod(e,m)
%o A353010                 yield d
%o A353010         p, q = q, nextprime(q)
%o A353010 A353010_list = list(islice(A353010_gen(),40)) # _Chai Wah Wu_, Jun 09 2022
%Y A353010 Cf. A276710.
%K A353010 nonn
%O A353010 1,3
%A A353010 _Hagen von Eitzen_, Apr 15 2022