A097035
Initial values for the iteration of the function f(x) = A063919(x) such that the iteration ends in a 5-cycle, i.e., in A097024.
Original entry on oeis.org
570, 870, 1230, 1290, 1326, 1482, 1530, 1686, 1698, 1710, 1794, 1866, 1878, 1890, 2058, 2070, 2142, 2154, 2166, 2178, 2238, 2250, 2502, 2802, 2814, 3042, 3222, 3630, 3702, 3714, 3726, 4350, 4494, 4506, 4518, 4914, 5010, 5142, 5154, 5166, 5284, 5418
Offset: 1
n = 570: list = {570, 870, 1290, [1878, 1890, 2142, 2178, 1482], 1878}; after 3 transients, a 5-cycle arises.
n = 1230: {1230, 1794, 2238, 2250, 1530, 1710, [1890, 2142, 2178, 1482, 1878]} ; the iteration to the 5-cycle is not necessarily monotone. - _Hartmut F. W. Hoft_, Jan 25 2024
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a063919[1] = 1; (* function a[] in A063919 by Jean-François Alcover *)
a063919[n_] := Total[Select[Divisors[n], GCD[#, n/#]==1&]]-n/;n>1
a097035Q[k_] := Module[{iter=NestWhileList[a063919, k, UnsameQ, All]}, Apply[Subtract, Reverse[Flatten[Position[iter, Last[iter]], 1]]]==5]
a097035[n_] := Select[Range[n], a097035Q]
a097035[5418] (* Hartmut F. W. Hoft, Jan 25 2024 *)
A127655
Numbers whose unitary aliquot sequences end in a unitary amicable pair, but which are not unitary amicable numbers themselves.
Original entry on oeis.org
102, 388, 436, 484, 812, 866, 1020, 1036, 1040, 1116, 1196, 1380, 1500, 1524, 1532, 1552, 1618, 1644, 1716, 1724, 1726, 1744, 1916, 2020, 2066, 2068, 2324, 2368, 2386, 2486, 2592, 2684, 2880, 2924, 3032, 3098, 3120, 3124, 3136, 3276, 3400, 3442, 3444, 3446, 3482
Offset: 1
a(5)=812 because the fifth non-unitary amicable number whose unitary aliquot sequence ends in a unitary amicable pair is 812.
- Riele, H. J. J. te; Unitary Aliquot Sequences. MR 139/72, Mathematisch Centrum, 1972, Amsterdam.
- Riele, H. J. J. te; Further Results On Unitary Aliquot Sequences. NW 2/73, Mathematisch Centrum, 1973, Amsterdam.
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UnitaryDivisors[n_Integer?Positive]:=Select[Divisors[n],GCD[ #,n/# ]==1&];sstar[n_]:=Plus@@UnitaryDivisors[n]-n;g[n_] := If[n > 0, sstar[n], 0];UnitaryTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];UnitaryAmicableNumberQ[k_]:=If[Nest[sstar,k,2]?k && !sstar[k]?k,True,False];Select[Range[2500],!UnitaryAmicableNumberQ[ # ] && UnitaryAmicableNumberQ[Last[UnitaryTrajectory[ # ]]] &]
A127652
Integers whose unitary aliquot sequences are longer than their ordinary aliquot sequences.
Original entry on oeis.org
25, 28, 36, 40, 50, 68, 70, 74, 94, 95, 98, 116, 119, 134, 142, 143, 154, 162, 170, 175, 182, 189, 190, 200, 220, 226, 242, 245, 262, 273
Offset: 1
a(5)=50 because the fifth integer whose unitary aliquot sequence is longer than its ordinary aliquot sequence is 50.
- Riele, H. J. J. te; Unitary Aliquot Sequences. MR 139/72, Mathematisch Centrum, 1972, Amsterdam.
- Riele, H. J. J. te; Further Results On Unitary Aliquot Sequences. NW 2/73, Mathematisch Centrum, 1973, Amsterdam.
Cf.
A127161,
A127162,
A127163,
A127164,
A063769,
A063990,
A097032,
A098007,
A097010,
A127653,
A098185,
A127654,
A063991,
A127655,
A097037,
A097036.
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UnitaryDivisors[n_Integer?Positive]:=Select[Divisors[n],GCD[ #,n/# ]==1&];sstar[n_]:=Plus@@UnitaryDivisors[n]-n;g[n_] := If[n > 0, sstar[n], 0];UnitaryTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];s[n_]:=DivisorSigma[1,n]-n;h[n_] := If[n > 0, s[n], 0];OrdinaryTrajectory[n_] := Most[NestWhileList[h, n, UnsameQ, All]];Select[Range[275],Length[UnitaryTrajectory[ # ]]>Length[OrdinaryTrajectory[ # ]] &]
A098186
If f[x]=(sum of unitary-proper divisors of x)=A063919[x] is iterated, the iteration may lead to a fixed point which is either 0 or belongs to A002827, a unitary-perfect-number >1: 6,60,90,87360... Sequence gives initial values for which the iteration ends in 87360, the 4th unitary perfect number.
Original entry on oeis.org
87360, 232608, 356640, 465144, 527712, 565728, 713208, 1018248, 1055352, 1211352, 1240032, 1303728, 1316904, 1352568, 1357584, 1360416, 1379280, 1550472, 1690440, 1835592, 2035608, 2078328, 2110632, 2262892, 2422632
Offset: 1
Iteration list started from n=1018248: {1018248, 1055352, 527712, 232608, 87360, 87360...}
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di[x_] :=Divisors[x];ta={{0}}; ud[x_] :=Part[di[x], Flatten[Position[GCD[di[x], Reverse[di[x]]], 1]]]; asu[x_] :=Apply[Plus, ud[x]]-x; nsf[x_, ho_] :=NestList[asu, x, ho] Do[g=n;s=Last[NestList[asu, n, 100]];If[Equal[s, 87360], Print[{n, s}]; ta=Append[ta, n]], {n, 1, 5000000}];ta = Delete[ta, 1]
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