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A205002 Least k such that n divides s(k)-s(j) for some j satisfying 1<=j where s(j)=j(j+1)/2.

%I A205002 #10 Sep 28 2018 10:30:29
%S A205002 2,2,3,4,3,5,4,8,4,6,6,5,7,5,6,16,9,6,10,6,8,7,12,9,7,8,7,11,15,8,16,
%T A205002 32,8,10,8,12,19,11,9,10,21,9,22,9,10,13,24,17,10,12,11,10,27,10,13,
%U A205002 11,12,16,30,11,31,17,11,64,11,17,34,12,14,13,36,12,37,20,12,13,12,18,40,18,13,22,42,14,13,23,17,13,45,13,16,15
%N A205002 Least k such that n divides s(k)-s(j) for some j satisfying 1<=j<k, where s(j)=j(j+1)/2.
%C A205002 See A204892 for a discussion and guide to related sequences.
%H A205002 Antti Karttunen, <a href="/A205002/b205002.txt">Table of n, a(n) for n = 1..11001</a>
%e A205002 (See example at A205001.)
%t A205002 s[n_] := s[n] = Binomial[n + 1, 2]; z1 = 500; z2 = 60;
%t A205002 Table[s[n], {n, 1, 30}]  (* A000217 *)
%t A205002 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205002 Table[u[m], {m, 1, z1}]  (* A193974 ? *)
%t A205002 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205002 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205002 d[n_] := d[n] = First[Delete[w[n], Position[w[n], 0]]]
%t A205002 Table[d[n], {n, 1, z2}]      (* A205001 *)
%t A205002 k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]
%t A205002 m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]
%t A205002 j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2
%t A205002 Table[k[n], {n, 1, z2}]      (* A205002 *)
%t A205002 Table[j[n], {n, 1, z2}]      (* A205003 *)
%t A205002 Table[s[k[n]], {n, 1, z2}]   (* A205004 *)
%t A205002 Table[s[j[n]], {n, 1, z2}]   (* A205005 *)
%t A205002 Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A205006 *)
%t A205002 Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A205007 *)
%o A205002 (PARI) A205002(n) = for(k=2,oo,my(sk=binomial(k+1,2)); for(j=1,k-1,if(!((sk-binomial(j+1,2))%n),return(k)))); \\ _Antti Karttunen_, Sep 27 2018
%Y A205002 Cf. A000217, A204892, A193974, A205006, A205007.
%K A205002 nonn
%O A205002 1,1
%A A205002 _Clark Kimberling_, Jan 21 2012
%E A205002 More terms from _Antti Karttunen_, Sep 27 2018