A069901 Smallest prime factor of n-th triangular number.

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, 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, 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

Offset: 1

Author

Reinhard Zumkeller, Apr 10 2002

Keywords

• nonn
• easy

Comments

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.

Examples

			A000217(10) = 10*(10+1)/2 = 55 = 5*11, therefore a(10) = 5.
		

Links

• Zak Seidov, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000217, A020639, A069902, A069903, A069904.

Programs

Mathematica

FactorInteger[#][[1,1]]&/@Accumulate[Range[100]] (* Harvey P. Dale, Apr 05 2014 *)

PARI

a(n) = if (n==1, 1, vecmin(factor(n*(n+1)/2)[,1]));

Formula

a(n) = A020639(A000217(n)).

a(4*k-1) = a(4*k) = 2.

From Zak Seidov, Jun 06 2013: (Start)

a(n) = 3 for n = {2, 5, 6, 9} + 12*k;

a(n) = 5 for n = {10, 25, 34, 49} + 60*k;

a(n) = 7 for n = {13, 97, 118, 133, 181, 202, 217, 238, 286, 301, 322, 406} + 420*k, etc. (End)

Extensions

Edited by N. J. A. Sloane, Sep 06 2008 at the suggestion of Franklin T. Adams-Watters