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A127662 Integers whose infinitary aliquot sequences end in an infinitary perfect number (A007357).

%I A127662 #18 Mar 11 2023 05:43:00
%S A127662 6,30,42,54,60,66,72,78,90,100,140,148,152,192,194,196,208,220,238,
%T A127662 244,252,268,274,292,296,298,300,336,348,350,360,364,372,374,380,382,
%U A127662 386,400,416,420,424,476,482,492,516,520,532,540,542,544,550,572,576,578,586
%N A127662 Integers whose infinitary aliquot sequences end in an infinitary perfect number (A007357).
%H A127662 Amiram Eldar, <a href="/A127662/b127662.txt">Table of n, a(n) for n = 1..75</a>
%H A127662 Graeme L. Cohen, <a href="http://dx.doi.org/10.1090/S0025-5718-1990-0993927-5">On an integer's infinitary divisors</a>, Math. Comp., 54 (1990), 395-411.
%H A127662 J. O. M. Pedersen, <a href="http://amicable.homepage.dk/tables.htm">Tables of Aliquot Cycles</a>. [Broken link]
%H A127662 J. O. M. Pedersen, <a href="http://web.archive.org/web/20140502102524/http://amicable.homepage.dk/tables.htm">Tables of Aliquot Cycles</a>. [Via Internet Archive Wayback-Machine]
%H A127662 J. O. M. Pedersen, <a href="/A063990/a063990.pdf">Tables of Aliquot Cycles</a>. [Cached copy, pdf file only]
%e A127662 a(5) = 60 because the fifth number whose infinitary aliquot sequence ends in an infinitary perfect number is 60.
%e A127662 6 -> 6 ...
%e A127662 30 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
%e A127662 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
%e A127662 54 -> 66 -> 78 -> 90 -> 90 -> ..
%e A127662 60 -> 60 -> ..
%e A127662 66 -> 78 -> 90 -> 90 -> ..
%e A127662 72 -> 78 -> 90 -> 90 -> ..
%e A127662 78 -> 90 -> 90 -> ..
%e A127662 90 -> 90 -> ..
%e A127662 100 -> 30 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
%e A127662 102 -> 114 -> 126 -> 114 -> ..  cycle but not in the sequence
%e A127662 114 -> 126 -> 114 -> .. cycle but not in the sequence
%e A127662 126 -> 114 -> 126 -> ..
%e A127662 140 -> 100 -> 30 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
%e A127662 148 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
%e A127662 152 -> 148 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
%e A127662 192 -> 148 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
%e A127662 194 -> 100 -> 30 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
%e A127662 196 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
%e A127662 208 -> 30 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
%e A127662 210 -> 366 -> 378 -> 582 -> 594 -> 846 -> 594 -> ..
%e A127662 220 -> 140 -> 100 -> 30 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
%e A127662 238 -> 194 -> 100 -> 30 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> ..
%e A127662 244 -> 66 -> 78 -> 90 -> 90 -> ..
%e A127662 246 -> 258 -> 270 -> 450 -> 330 -> 534 -> 546 -> 798 -> 1122 -> 1470 -> 2130 -> 3054 -> 3066 -> 4038 -> 4050 -> 2346 -> 2838 -> 3498 -> 4278 -> 4938 -> 4950 -> 4410 -> 4590 -> 8370 -> 14670 -> 14850 -> 22590 -> 22770 -> 29070 -> 35730 -> 35910 -> 79290 -> 79470 -> 79650 -> 107550 -> 79650 -> ..
%p A127662 isA007357 := proc(n)
%p A127662     A049417(n) = 2*n ;
%p A127662     simplify(%) ;
%p A127662 end proc:
%p A127662 isA127662 := proc(n)
%p A127662     local trac,x;
%p A127662     x := n ;
%p A127662     trac := [x] ;
%p A127662     while true do
%p A127662         x := A049417(x)-trac[-1] ;
%p A127662         if x = 0 then
%p A127662             return false ;
%p A127662         elif x in trac then
%p A127662             return isA007357(x) ;
%p A127662         end if;
%p A127662         trac := [op(trac),x] ;
%p A127662     end do:
%p A127662 end proc:
%p A127662 for n from 1 do
%p A127662     if isA127662(n) then
%p A127662         printf("%d,\n",n) ;
%p A127662     end if;
%p A127662 end do: # _R. J. Mathar_, Oct 05 2017
%t A127662 ExponentList[n_Integer,factors_List]:={#,IntegerExponent[n,# ]}&/@factors;InfinitaryDivisors[1]:={1}; InfinitaryDivisors[n_Integer?Positive]:=Module[ { factors=First/@FactorInteger[n], d=Divisors[n] }, d[[Flatten[Position[ Transpose[ Thread[Function[{f,g}, BitOr[f,g]==g][ #,Last[ # ]]]&/@ Transpose[Last/@ExponentList[ #,factors]&/@d]],_?(And@@#&),{1}]] ]] ] Null;properinfinitarydivisorsum[k_]:=Plus@@InfinitaryDivisors[k]-k;g[n_] := If[n > 0,properinfinitarydivisorsum[n], 0];iTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];InfinitaryPerfectNumberQ[0]=False;InfinitaryPerfectNumberQ[k_Integer] :=If[properinfinitarydivisorsum[k]==k,True,False];Select[Range[500],InfinitaryPerfectNumberQ[Last[iTrajectory[ # ]]] &]
%t A127662 s[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - n; s[0] = s[1] = 0; q[n_] := Module[{v = NestWhileList[s, n, UnsameQ, All]}, v[[-1]] != n && v[[-2]] == v[[-1]] > 0]; Select[Range[3200], q] (* _Amiram Eldar_, Mar 11 2023 *)
%Y A127662 Cf. A007357, A126168, A127661 - A127667.
%K A127662 hard,nonn
%O A127662 1,1
%A A127662 _Ant King_, Jan 26 2007
%E A127662 More terms from _Amiram Eldar_, Mar 11 2023