A091321 OU-Sigma multiperfect numbers.

1, 6, 28, 90, 120, 496, 672, 8128, 10080, 63700, 220500, 523776, 1323000, 1528800, 2056320, 7856640, 33550336, 44553600, 162729000, 252927360, 459818240, 1379454720, 1476304896, 1980840960, 8589869056

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.

Links

Table of n, a(n) for n=1..25.

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

Cf. A107749, A091322.

Sequence in context: A302650 A055711 A141255 * A125310 A336535 A342380

Adjacent sequences: A091318 A091319 A091320 * A091322 A091323 A091324

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