A164978 - OEIS

%I #13 Aug 05 2019 18:03:38

%S 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,

%T 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,

%U 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

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

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

%H Alois P. Heinz and T. D. Noe, <a href="/A164978/b164978.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)

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

%e 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}.

%e 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.

%p with(numtheory):

%p a:= n-> nops(select(x-> x>=n, divisors(n*(n+1)/2))):

%p seq(a(n), n=1..120);

%Y Cf. A164977, A035470.

%K easy,nonn

%O 1,3

%A _Alois P. Heinz_, Sep 03 2009