A093863 Unitary sigma-unitary phi super perfect numbers: USUP(USUP(n))= n/k for some integer k.

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 17, 18, 20, 24, 34, 36, 40, 48, 68, 72, 80, 136, 144, 256, 257, 272, 514, 768, 1028, 1280, 2056, 2304, 2808, 4112, 4320, 4352, 20280, 65536, 65537, 65792, 88704, 131074, 196416, 196608, 262148, 327680, 524296, 589824, 998400

Offset: 1

Author

Yasutoshi Kohmoto, May 11 2004

Keywords

• nonn

Comments

USUP(.)= A109712(.). Where k values are 1, they define fixed points of the function USUP(USUP(n)). k values larger than 1 exist, for example USUP(USUP(4320))= 4320/2.

k = 2 for 4320, 20280, 88704, 196416, 998400, ... - Amiram Eldar, Mar 01 2019

Crossrefs

Cf. A092760, A109712.

Programs

Maple

for n from 1 to 20000 do if n mod A109712(A109712(n)) = 0 then printf("%d,",n); end if; end do:

Mathematica

usigma[1]=1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); A047994[n_] := Times @@ (Power @@@ FactorInteger[n] - 1); A006519[n_] := 2^IntegerExponent[ n, 2]; usup[1] = 1; usup[n_ /; IntegerQ[Log[2, n]]] := n+1; usup[n_] := usigma[ A006519[n] ]*A047994[ n/A006519[n] ];  aQ[n_]:=Divisible[n,usup[usup[n]]]; Select[Range[10000], aQ] (* Amiram Eldar, Mar 01 2019 after Jean-François Alcover at A109712 *)

Extensions

More terms from Amiram Eldar, Mar 01 2019