OFFSET
1,1
COMMENTS
Since s = 0 if k is odd, the number k is necessarily even and then the number of even divisors s is always a multiple of the number of odd divisors r. Note that t = r + s may not be a divisor even if both r and s are divisors. For example, if k = 144, then r = 3, s = 12, but t = r + s = 15.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = k is even, r = number of odd divisors of k, s = number of even divisors of k and t = r + s = number of divisors of k, are all divisors of k.
EXAMPLE
a(3) = 18 since r = 3, s = 3 and t = r + s = 6 are all divisors.
MAPLE
with(numtheory); A:=[]: for w to 1 do for k from 1 to 5000 do n:=2*k; S:=divisors(n); r:=nops( select(z->type(z, odd), S) ); s:=nops( select(z->type(z, even), S) ); t:=r+s; if andmap(z -> n mod z = 0, [r, s, t]) then A:=[op(A), n]; print(n, r, s, t); fi; od od; A;
MATHEMATICA
oddtau[n_] := DivisorSigma[0, n/2^IntegerExponent[n, 2]]; seqQ[n_] := Module[{d = DivisorSigma[0, n], o = odd[n]}, Divisible[n, d] && Divisible[n, o] && Divisible[n, d - o]]; Select[Range[2, 2000, 2], seqQ] (* Amiram Eldar, Jan 15 2020 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Walter Kehowski, Jun 24 2006
EXTENSIONS
Offset corrected by Amiram Eldar, Jan 15 2020
STATUS
approved