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A093863 Unitary sigma-unitary phi super perfect numbers: USUP(USUP(n))= n/k for some integer k. 0

%I #11 Mar 02 2019 04:26:10

%S 1,2,3,4,5,6,8,9,10,12,16,17,18,20,24,34,36,40,48,68,72,80,136,144,

%T 256,257,272,514,768,1028,1280,2056,2304,2808,4112,4320,4352,20280,

%U 65536,65537,65792,88704,131074,196416,196608,262148,327680,524296,589824,998400

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

%C 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.

%C k = 2 for 4320, 20280, 88704, 196416, 998400, ... - _Amiram Eldar_, Mar 01 2019

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

%t 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 *)

%Y Cf. A092760, A109712.

%K nonn

%O 1,2

%A _Yasutoshi Kohmoto_, May 11 2004

%E More terms from _Amiram Eldar_, Mar 01 2019

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)