cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A127653 Integers whose unitary aliquot sequences terminate in 0, including 1 but excluding the other trivial cases in which n is itself either a prime or a prime power.

Original entry on oeis.org

1, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 34, 35, 36, 38, 39, 40, 44, 45, 46, 48, 50, 51, 52, 55, 56, 57, 58, 62, 63, 65, 68, 69, 70, 72, 74, 75, 76, 77, 80, 82, 84, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 104, 105, 106, 108, 110, 111, 112, 115
Offset: 1

Views

Author

Ant King, Jan 24 2007

Keywords

Examples

			a(5) = 15 because the fifth integer that is neither prime nor a prime power and whose unitary aliquot sequence terminates in 0 is 15.

Links

Crossrefs

Cf. A127652, A097032, A097010, A098185, A127654, A063991, A097037, A097036.

Programs

  • Mathematica
    UnitaryDivisors[n_Integer?Positive] := Select[Divisors[n], GCD[ #, n/# ] == 1 \ &]; sstar[n_] := Plus @@ UnitaryDivisors[ n] - n; pp[k_] := If[Length[ FactorInteger[k]] == 1, True, False]; g[n_] := If[n > 0, sstar[n], 0]; UnitaryTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]]; Select[Range[100], Last[UnitaryTrajectory[ # ]] == 0 && ! pp[ # ] &]
s[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - n; s[0] = s[1] = 0; q[n_] := If[PrimeNu[n] == 1, False, Module[{v = NestWhileList[s, n, UnsameQ, All]}, v[[-1]] == 0]]; Select[Range[120], q] (* Amiram Eldar, Mar 11 2023 *)

Extensions

More terms from Amiram Eldar, Mar 11 2023

A127654 Unitary aspiring numbers.

Original entry on oeis.org

66, 78, 244, 292, 476, 482, 578, 648, 680, 688, 770, 784, 832, 864, 956, 958, 976, 1168, 1354, 1360, 1392, 1488, 1600, 1658, 1670, 1906, 2232, 2264, 2294, 2376, 2480, 2552, 2572, 2576, 2626, 2712, 2732, 2806, 2842, 2870, 2904, 2912, 2992, 3024, 3096, 3140, 3172
Offset: 1

Views

Author

Ant King, Jan 24 2007

Keywords

Comments

A unitary aspiring number is an integer whose unitary aliquot sequences ends by meeting a unitary-perfect number (A098185) in its trajectory, but is not unitary-perfect itself. There are 1693 such numbers <=100000 and of these 82860 and 97020 generate the longest unitary aliquot sequences (according to A097032), each having length 18 and ending with the unitary perfect number 90.

Examples

			a(5) = 476 because the fifth non-unitary-perfect number whose unitary aliquot sequence ends in a unitary-perfect number is 476.

Links

Crossrefs

Cf. A002827, A097032, A127652, A097010, A098185, A127653, A063991, A097037, A097036.

Programs

  • Mathematica
    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]]; UnitaryPerfectNumberQ[0] = 0; UnitaryPerfectNumberQ[k_] := If[sstar[k] == k, True, False]; UnitaryAspiringNumberQ[k_] := If[UnitaryPerfectNumberQ[Last[ UnitaryTrajectory[k]]] && ! UnitaryPerfectNumberQ[k], True, False]; Select[Range[2500], UnitaryAspiringNumberQ[ # ] &]
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

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

Views

Author

Ant King, Jan 25 2007

Keywords

Examples

			a(5)=812 because the fifth non-unitary amicable number whose unitary aliquot sequence ends in a unitary amicable pair is 812.

References

  • 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.

Links

Crossrefs

Cf. A063991, A097037, A127654, A098185, A097032, A097010, A127653, A097036.

Programs

  • Mathematica
    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[ # ]]] &]

Extensions

More terms from Amiram Eldar, Apr 06 2019

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

Views

Author

Ant King, Jan 24 2007

Keywords

Comments

Here the length of an aliquot sequence is defined to be the length of the transient part of its trajectory + the length of its terminal cycle.

Examples

			a(5)=50 because the fifth integer whose unitary aliquot sequence is longer than its ordinary aliquot sequence is 50.

References

  • 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.

Links

Crossrefs

Cf. A127161, A127162, A127163, A127164, A063769, A063990, A097032, A098007, A097010, A127653, A098185, A127654, A063991, A127655, A097037, A097036.

Programs

  • Mathematica
    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[ # ]] &]

Formula

Sequence gives those values of n for which A097032(n)>A098007(n).

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

Views

Author

Labos Elemer, Aug 31 2004

Keywords

Examples

			Iteration list started from n=1018248: {1018248, 1055352, 527712, 232608, 87360, 87360...}

Crossrefs

Cf. A063919, A002827, A063991, A097024, A097030-A097037, A098185.

Programs

  • Mathematica
    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]

A098188 Irregular triangle with 4 columns which contains in each row the members of a 4-cycle under the map x->A063919(x), iteration of summing the proper-unitary divisors.

Original entry on oeis.org

263820, 263940, 280380, 280500, 395730, 395910, 420570, 420750, 172459210, 218725430, 272130250, 218628662, 209524210, 246667790, 231439570, 230143790, 384121920, 384296640, 408233280, 408408000
Offset: 1

Views

Author

Labos Elemer, Sep 02 2004

Keywords

Comments

Initial values attracted by this sequence are in A098187.
The iteration of this function also contains 2-cycles like 114->126->114... or 1140 -> 1260 ->1140,... or 3-cycles like 30->42->54->30->....

Examples

			The first line represents the 4-cycle  280500->263820->263940->280380->280500->...,
The second line represents the 4-cycle 420750->395730->395910->420570->420750->..

Links

Crossrefs

Cf. A098185, A098186, A002827, A097024, A097030, A098187.
Cf. A319902 (where the terms are entered by increasing value).

Extensions

More terms from Michel Marcus, Oct 05 2018

A098187 Initial seeds x which will enter a cycle of length 4 under the iteration of x -> A063919(x), the sum of proper unitary divisors.

Original entry on oeis.org

81570, 114270, 137046, 169998, 177906, 182082, 182094, 185190, 194574, 194586, 201642, 203442, 204420, 204540, 212466, 212874, 213870, 219306, 219318, 230874, 231438, 231834, 231846, 232626, 237678, 238134, 242634, 258882, 259338, 259350
Offset: 1

Views

Author

Labos Elemer, Sep 02 2004

Keywords

Comments

The sequence is the attractor-basin of set of {C4} cycles belonging to this iteration.
The {C4} attractor-set is displayed separately in A098188.

Examples

			81570 is in the sequence because its track under the iterated map is 81570, 114270, 182082, 182094, 232626, 237678, 305682, 352878, 360978, 403662, 420738, [420750, 395730, 395910, 420570], 420750.., where the cycle is indicated by brackets. The 4 recurrent terms appear after 11 transients for this case.

Crossrefs

Cf. A098185, A098186, A002827, A097024, A097030.
Showing 1-7 of 7 results.

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