OFFSET
1,2
COMMENTS
Two subsets of terms:
1) If 2^p - 1 is a Mersenne prime (p is in A000043 and 2^p-1 is in A000668), then m = 2^(p-1) is a term that belongs to A019279: the even superperfect numbers (2, 4, 16, 64, 4096, ...). Proof: sigma(m) = 1+2+...+2^(p-1) = 2^p - 1 that is a Mersenne prime so tau(2^p-1) = 2 that divides m = 2^(p-1); indeed, m/tau(sigma(m)) = 2^(p-2).
2) If m = 2^(p-1) is a term as above, then 3*m is another term (see example) with 3*m/tau(sigma(3*m)) = 2^(p-2).
EXAMPLE
48 = 2^4 * 3, so, sigma(48) = sigma(2^4) * sigma(3) = (2^5 - 1) * (1+3) = 31 * 4 = 124; then, tau(2^2 * 31) = tau(4) * tau(31) = 3 * 2 = 6, and 48/6 = 8 = 2^3, hence 48 is a term.
MAPLE
with(numtheory) filter:= m -> m/tau(sigma(m)) = floor(m/tau(sigma(m))) : select(filter, [$1..650]);
MATHEMATICA
Select[Range[630], Divisible[#, DivisorSigma[0, DivisorSigma[1, #]]] &] (* Amiram Eldar, Jul 30 2020 *)
PROG
(PARI) isok(m) = !(m % numdiv(sigma(m))); \\ Michel Marcus, Jul 30 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Bernard Schott, Jul 29 2020
STATUS
approved