A091321 - OEIS

%I #16 Mar 17 2019 10:33:40

%S 1,6,28,90,120,496,672,8128,10080,63700,220500,523776,1323000,1528800,

%T 2056320,7856640,33550336,44553600,162729000,252927360,459818240,

%U 1379454720,1476304896,1980840960,8589869056

%N OU-Sigma multiperfect numbers.

%C The OU-Sigma function is defined as OU-Sigma(n) = A107749(n).

%C Then an OU-Sigma perfect number satisfies OU-Sigma(n) = k*n for some k.

%C 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.

%C Also in the sequence are 33550336, 8589869056, 22144573440, 51001180160, 153003540480, 243643438080, 583125903360, 71724486113280, 1555825650042470400, but there may be missing terms in between.

%e 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, ...

%t 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 *)

%o (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

%o isok(n) = (f(n) % n) == 0; \\ _Michel Marcus_, Jan 24 2019

%Y Cf. A107749, A091322.

%K nonn,more

%O 1,2

%A _Yasutoshi Kohmoto_, Feb 17 2004

%E Terms 220500 to 2056320 by _R. J. Mathar_, Jun 02 2011

%E Corrected and extended by _Michel Marcus_, Jan 24 2019

%E a(19)-a(25) from _Amiram Eldar_, Mar 17 2019