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

1,2

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.

Links

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

Formula

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

a(4k-1) = a(4k) = 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)

Example

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A350285 A347381 A075392 * A115039 A032536 A115061

Adjacent sequences: A069898 A069899 A069900 * A069902 A069903 A069904

Keyword

nonn

Author

Reinhard Zumkeller, Apr 10 2002

Extensions

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

Status

approved