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 A127662 Integers whose infinitary aliquot sequences end in an infinitary perfect number (A007357). 4
 6, 30, 42, 54, 60, 66, 72, 78, 90, 100, 140, 148, 152, 192, 194, 196, 208, 220, 238, 244, 252, 268, 274, 292, 296, 298, 300, 336, 348, 350, 360, 364, 372, 374, 380, 382, 386, 400, 416, 420, 424, 476, 482, 492 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 LINKS Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp., 54 (1990), 395-411. J. O. M. Pedersen, Tables of Aliquot Cycles [Broken link] J. O. M. Pedersen, Tables of Aliquot Cycles [Via Internet Archive Wayback-Machine] J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only] EXAMPLE a(5)=60 because the fifth number whose infinitary aliquot sequence ends in an infinitary perfect number is 60. 6 -> 6 ... 30 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> .. 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> .. 54 -> 66 -> 78 -> 90 -> 90 -> .. 60 -> 60 -> .. 66 -> 78 -> 90 -> 90 -> .. 72 -> 78 -> 90 -> 90 -> .. 78 -> 90 -> 90 -> .. 90 -> 90 -> .. 100 -> 30 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> .. 102 -> 114 -> 126 -> 114 -> ..  cycle but not in the sequence 114 -> 126 -> 114 -> .. cycle but not in the sequence 126 -> 114 -> 126 -> .. 140 -> 100 -> 30 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> .. 148 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> .. 152 -> 148 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> .. 192 -> 148 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> .. 194 -> 100 -> 30 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> .. 196 -> 54 -> 66 -> 78 -> 90 -> 90 -> .. 208 -> 30 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> .. 210 -> 366 -> 378 -> 582 -> 594 -> 846 -> 594 -> .. 220 -> 140 -> 100 -> 30 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> .. 238 -> 194 -> 100 -> 30 -> 42 -> 54 -> 66 -> 78 -> 90 -> 90 -> .. 244 -> 66 -> 78 -> 90 -> 90 -> .. 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 -> .. MAPLE isA007357 := proc(n)     A049417(n) = 2*n ;     simplify(%) ; end proc: isA127662 := proc(n)     local trac, x;     x := n ;     trac := [x] ;     while true do         x := A049417(x)-trac[-1] ;         if x = 0 then             return false ;         elif x in trac then             return isA007357(x) ;         end if;         trac := [op(trac), x] ;     end do: end proc: for n from 1 do     if isA127662(n) then         printf("%d, \n", n) ;     end if; end do: # R. J. Mathar, Oct 05 2017 MATHEMATICA 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[ # ]]] &] CROSSREFS Cf. A007357, A126168, A127661 - A127667. Sequence in context: A002445 A151711 A130512 * A003062 A327157 A309312 Adjacent sequences:  A127659 A127660 A127661 * A127663 A127664 A127665 KEYWORD hard,nonn AUTHOR Ant King, Jan 26 2007 STATUS approved

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Last modified December 7 03:27 EST 2019. Contains 329836 sequences. (Running on oeis4.)