A093493 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.

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, 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, 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

Offset

1,7

Comments

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.

Does every nonnegative integer occur in this sequence? - Franklin T. Adams-Watters, May 12 2006

Links

Table of n, a(n) for n=1..83.

Example

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.

Prog

(PARI) a(n) = sum(r=1, n-1, numdiv(n-r)==numdiv(n+r)) \\ Jason Yuen, Aug 27 2024

Crossrefs

Cf. A093488, A093491, A093492, A093494.

Sequence in context: A354143 A358616 A366105 * A087162 A288914 A046925

Adjacent sequences: A093490 A093491 A093492 * A093494 A093495 A093496

Keyword

nonn

Author

Amarnath Murthy, Apr 16 2004

Extensions

More terms from Franklin T. Adams-Watters, May 12 2006

Status

approved