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A127662 Integers whose infinitary aliquot sequences end in an infinitary perfect number (A007357). 5
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 (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[ # ]]] &]
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 *)
CROSSREFS
Sequence in context: A002445 A151711 A130512 * A003062 A327157 A336216
KEYWORD
hard,nonn
AUTHOR
Ant King, Jan 26 2007
EXTENSIONS
More terms from Amiram Eldar, Mar 11 2023
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)