OFFSET
1,2
COMMENTS
The OU-Sigma function is defined as OU-Sigma(n) = A107749(n).
Then an OU-Sigma perfect number satisfies OU-Sigma(n) = k*n for some k.
Every perfect number is here because OE-Sigma(2^(m-1)*M_m) = Sigma(2^(m-1))*UnitarySigma(M_m) = Sigma(2^(m-1))*Sigma(M_m) = 2^m*M_m.
Also in the sequence are 33550336, 8589869056, 22144573440, 51001180160, 153003540480, 243643438080, 583125903360, 71724486113280, 1555825650042470400, but there may be missing terms in between.
EXAMPLE
Sequence begins 2*3, 2*3^2*5, 2^2*7, 2^2*5^2*7^2*13, 2^3*3*5, 2^4*31, 2^5*3^2*5*7, ...
MATHEMATICA
fun[p_, e_] := If[p==2, 2^(e+1)-1, p^e+1]; f[n_] := If[n==1, 1, Times @@ fun @@@ FactorInteger[n]]; aQ[n_] := Divisible[f[n], n]; Select[Range[65000], aQ] (* Amiram Eldar, Mar 17 2019 *)
PROG
(PARI) f(n)= my(fm=factor(n)); prod(k=1, matsize(fm)[1], if(fm[k, 1]==2, 2^(fm[k, 2]+1)-1, fm[k, 1]^fm[k, 2]+1)); \\ A107749
isok(n) = (f(n) % n) == 0; \\ Michel Marcus, Jan 24 2019
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Yasutoshi Kohmoto, Feb 17 2004
EXTENSIONS
Terms 220500 to 2056320 by R. J. Mathar, Jun 02 2011
Corrected and extended by Michel Marcus, Jan 24 2019
a(19)-a(25) from Amiram Eldar, Mar 17 2019
STATUS
approved