A218767 Total number of divisors and anti-divisors of n.

1, 2, 3, 4, 4, 5, 5, 6, 5, 7, 5, 8, 6, 7, 7, 7, 7, 10, 5, 9, 7, 9, 7, 10, 8, 7, 9, 11, 5, 11, 7, 12, 9, 7, 9, 11, 7, 11, 9, 12, 6, 13, 7, 9, 13, 9, 7, 13, 9, 12, 7, 13, 9, 11, 9, 11, 9, 11, 9, 18, 6, 9, 13, 9, 9, 13, 11, 13, 7, 13, 7, 18, 9, 9, 11, 11, 13, 13, 5, 15, 11, 11, 9, 16, 12, 9

Offset: 1

Author

Juri-Stepan Gerasimov, Feb 05 2013

Keywords

• nonn

Comments

Or tau(n) + anti-tau(n), where anti-tau = A066272.

Total sum of divisors and anti-divisors of n or sigma(n) + A066417(n): 1, 3, 6, 10, 11, 16, 18, 23, 21, 32, 24, 41, 33, 40, 42, 45, 46, 67, 38, 66, 54, 72, 58, 83, 70, 66, 82, 102, 54, 108,...

Numbers n such that sigma(n) = n + anti-sigma(n): A074751.

Numbers n such that Chowla's function(n) = anti-sigma(n): 1, 2, 16, 60, 72,...

Number of divisors of n minus number of anti-divisors of n or tau(n) - anti-tau(n): 1, 2, 1, 2, 0, 3, -1, 2, 1, 1, -1, 4, -2, 1, 1, 3, -3, 2, -1, 3, 1, -1, -3, 6, -2, 1, -1, 1, -1, 5, -3, 0, -1, 1, -1, 7, -3, -3, -1, 4, -2, 3, -3, 3, -1,...

Product of number of divisors of n and number of anti-divisors of n, or tau(n)*anti-tau(n): 0, 0, 2, 3, 4, 4, 6, 8, 6, 12, 6, 12, 8, 12, 12, 10, 10, 24, 6, 18, 12, 20, 10, 16, 15, 12, 20, 30, 6, 24,...

Number of ways to write n as k*(k - m) with k divisor and m anti-divisor of n: 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0,...

Numbers which are not of the form k*(k - m), k divisor, m anti-divisor (i.e., where the number of ways is zero): 1, 2, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 19, 21, 23, 24, 25, 26, 29,

Crossrefs

Cf. A000005, A000203, A001065, A066272, A066417, A027750, A048050, A130799.

Programs

Maple

A218767 := proc(n)
        numtheory[tau](n)+A066272(n) ;
end proc: # R. J. Mathar, Feb 16 2013

Formula

a(n) = A000005(n) + A066272(n).