A164978 Number of divisors of n*(n+1)/2 that are >= n.

1, 1, 2, 2, 2, 2, 3, 4, 3, 2, 4, 4, 2, 4, 7, 4, 3, 3, 4, 7, 4, 2, 6, 8, 3, 4, 7, 4, 4, 4, 5, 9, 4, 4, 11, 6, 2, 4, 11, 6, 4, 4, 4, 11, 6, 2, 8, 11, 4, 6, 7, 4, 4, 7, 11, 11, 4, 2, 8, 8, 2, 6, 16, 11, 7, 4, 4, 7, 8, 4, 9, 9, 2, 6, 11, 8, 8, 4, 8, 18, 5, 2, 8, 15, 4, 4, 11, 6, 6, 11, 8, 7, 4, 4, 18, 10, 3, 8

Offset

1,3

Comments

a(n) = 2 <=> the set S = {1..n} has only one decomposition into smaller subsets with equal element sum.

Links

Alois P. Heinz and T. D. Noe, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Formula

a(n) = |{d|n*(n+1)/2 : d>=n}|.

Example

a(6) = 2, because 6*7/2 = 21 with divisors {1,3,7,21}, but only 7 and 21 are >= 6. S = {1..6} has only one decomposition into smaller subsets with equal element sum: {1,6}, {2,5}, {3,4}.
a(7) = 3; 7*8/2 = 28 with divisors {1,2,4,7,14,28}, 3 of which are >= 7. S = {1..7} has 5 decompositions into smaller subsets with equal element sum.

Maple

with(numtheory):
a:= n-> nops(select(x-> x>=n, divisors(n*(n+1)/2))):
seq(a(n), n=1..120);

Mathematica

a[n_] := DivisorSum[n*(n+1)/2, 1 &, # >= n &]; Array[a, 100] (* Amiram Eldar, Mar 09 2026 *)

Prog

(PARI) a(n) = sumdiv(n*(n+1)/2, d, d >= n); \\ Amiram Eldar, Mar 09 2026

Crossrefs

Cf. A164977, A035470.

Sequence in context: A262944 A322789 A069904 * A119789 A025424 A216505

Adjacent sequences: A164975 A164976 A164977 * A164979 A164980 A164981

Keyword

easy,nonn

Author

Alois P. Heinz, Sep 03 2009

Status

approved