A093493 - OEIS

%I #9 Aug 27 2024 21:39:39

%S 0,0,0,1,1,1,2,3,3,3,3,4,2,3,5,5,5,7,2,6,6,6,5,11,6,6,9,7,4,12,5,10,

%T 10,7,10,16,6,8,11,11,8,17,8,10,15,10,10,20,6,14,13,13,9,21,12,18,13,

%U 13,11,29,7,12,20,16,14,21,13,14,13,16,18,33,13,16,23,16,16,28,13,24,20,15,16

%N Define the total divisor symmetry of a number n to be the number of values r takes such that n-r and n+r have the same number of divisors. Sequence contains the total divisor symmetry of n.

%C Number of partitions of 2n in two parts with equal number divisors. Conjecture: (1) No term is zero for n > 3. (2) Every number k appears finitely many times in the sequence. i.e. for every k there exists a number f(k) so that for all n > f(k), a(n) > k. Subsidiary sequences: (1) The frequency of n. (2) The greatest number m so that a(m) = n.

%C Does every nonnegative integer occur in this sequence? - _Franklin T. Adams-Watters_, May 12 2006

%e a(15) = 5 and the values r takes are 2,3,4,7 and 8 giving the number pairs (13,17), (12,18), (11,19), (8,22) and (7,23) with same number of divisors.

%o (PARI) a(n) = sum(r=1,n-1,numdiv(n-r)==numdiv(n+r)) \\ _Jason Yuen_, Aug 27 2024

%Y Cf. A093488, A093491, A093492, A093494.

%K nonn

%O 1,7

%A _Amarnath Murthy_, Apr 16 2004

%E More terms from _Franklin T. Adams-Watters_, May 12 2006