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

Author

Yasutoshi Kohmoto, Feb 17 2004

Keywords

• nonn
• more

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.

Examples

			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, ...
		

Crossrefs

Cf. A107749, A091322.

Programs

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 *)

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

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