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A069901 Smallest prime factor of n-th triangular number.

%I A069901 #22 Aug 14 2025 20:23:27
%S A069901 1,3,2,2,3,3,2,2,3,5,2,2,7,3,2,2,3,3,2,2,3,11,2,2,5,3,2,2,3,3,2,2,3,5,
%T A069901 2,2,19,3,2,2,3,3,2,2,3,23,2,2,5,3,2,2,3,3,2,2,3,29,2,2,31,3,2,2,3,3,
%U A069901 2,2,3,5,2,2,37,3,2,2,3,3,2,2,3,41,2,2,5,3,2,2,3,3,2,2,3,5,2,2,7,3,2
%N A069901 Smallest prime factor of n-th triangular number.
%C A069901 Or, a(1) = 1, then the smallest nontrivial k (>1) which divides the sum of (next n) numbers from k+1 to k+n or smallest k > 1 that divides nk + n(n+1)/2. - _Amarnath Murthy_, Sep 22 2002. For example, a(7) = 4, which is the smallest nontrivial number that divides the sum 5+6+...+11, of 7 numbers.
%H A069901 Zak Seidov, <a href="/A069901/b069901.txt">Table of n, a(n) for n = 1..10000</a>
%F A069901 a(n) = A020639(A000217(n)).
%F A069901 a(4*k-1) = a(4*k) = 2.
%F A069901 From _Zak Seidov_, Jun 06 2013: (Start)
%F A069901 a(n) = 3 for n = {2, 5, 6, 9} + 12*k;
%F A069901 a(n) = 5 for n = {10, 25, 34, 49} + 60*k;
%F A069901 a(n) = 7 for n = {13, 97, 118, 133, 181, 202, 217, 238, 286, 301, 322, 406} + 420*k, etc. (End)
%e A069901 A000217(10) = 10*(10+1)/2 = 55 = 5*11, therefore a(10) = 5.
%t A069901 FactorInteger[#][[1,1]]&/@Accumulate[Range[100]] (* _Harvey P. Dale_, Apr 05 2014 *)
%o A069901 (PARI) a(n) = if (n==1, 1, vecmin(factor(n*(n+1)/2)[,1]));
%Y A069901 Cf. A000217, A020639, A069902, A069903, A069904.
%K A069901 nonn,easy
%O A069901 1,2
%A A069901 _Reinhard Zumkeller_, Apr 10 2002
%E A069901 Edited by _N. J. A. Sloane_, Sep 06 2008 at the suggestion of _Franklin T. Adams-Watters_