cp's OEIS Frontend

A387064 Total number of entries in rows 0 to n of Pascal's triangle multiple of n.

%I A387064 #47 Aug 22 2025 20:21:40
%S A387064 0,3,1,2,2,4,3,6,4,6,10,10,12,12,21,22,8,16,18,18,30,42,47,22,38,20,
%T A387064 74,18,65,28,81,30,16,113,136,132,94,36,147,195,140,40,162,42,199,210,
%U A387064 217,46,126,42,146,302,261,52,110,335,243,374,394,58,363,60,465,416
%N A387064 Total number of entries in rows 0 to n of Pascal's triangle multiple of n.
%H A387064 Chai Wah Wu, <a href="/A387064/b387064.txt">Table of n, a(n) for n = 0..10000</a>
%F A387064 a(p) = p-1, a(p^2) = p*(p-1) for p prime. Conjecture: a(p^k) = (p-1)*p^(k-1) for p prime. - _Chai Wah Wu_, Aug 21 2025
%e A387064 The first two rows of Pascal's triangle are [1] and [1, 1]. Since all elements are divisible by 1, a(1) equals the total number of such divisible terms: 1 + 2 = 3.
%t A387064 a[n_] := Sum[Boole[Divisible[Binomial[k, i], n]], {k, 0, n}, {i, 0, k}]; a[0] = 0; Array[a, 100, 0] (* _Amiram Eldar_, Aug 17 2025 *)
%o A387064 (PARI) a(n) = if (n, sum(r=0, n, sum(k=0, r, !(binomial(r,k) % n))), 0); \\ _Michel Marcus_, Aug 15 2025
%o A387064 (Python)
%o A387064 from sympy import isprime, integer_nthroot
%o A387064 def A387064(n):
%o A387064     if isprime(n): return n-1
%o A387064     a, b = integer_nthroot(n,2)
%o A387064     if b and isprime(a): return n-a
%o A387064     r, c = [1], n==1
%o A387064     for m in range(n):
%o A387064         s = [1]
%o A387064         for i in range(m):
%o A387064             s.append((r[i]+r[i+1])%n)
%o A387064             c += s[-1]==0
%o A387064         r = s+[1]
%o A387064         c += (n==1)<<1
%o A387064     return int(c) # _Chai Wah Wu_, Aug 21 2025
%Y A387064 Cf. A001316, A006046, A006047, A006048, A007318, A194458, A194459, A382720-A382731, A386953.
%K A387064 nonn,new
%O A387064 0,2
%A A387064 _Jean-Marc Rebert_, Aug 15 2025