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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

Table of n, a(n) for n=1..44.

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.)