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

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, 516, 520, 532, 540, 542, 544, 550, 572, 576, 578, 586

Offset: 1

Author

Ant King, Jan 26 2007

Keywords

• hard
• nonn

Examples

			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 -> ..
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..75
• 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]

Crossrefs

Cf. A007357, A126168, A127661 - A127667.

Programs

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[ # ]]] &]
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 *)

Extensions

More terms from Amiram Eldar, Mar 11 2023