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 1-5 of 5 results.

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).

A290143 Numbers n such that transient part of the unitary aliquot sequence for n sets a new record.

Original entry on oeis.org

1, 2, 10, 14, 22, 38, 70, 134, 138, 170, 190, 210, 318, 426, 1398, 4170, 6870, 8454, 19866, 22470, 36282, 38370, 70770, 84774, 98790, 132990, 474642, 705990, 961650
Offset: 1

Views

Author

Amiram Eldar, Jul 21 2017

Keywords

Comments

The unitary version of A098009.
The record values are in A290144.

Examples

			The unitary aliquot sequence of 134 is: 134, 70, 74, 40, 14, 10, 8, 1. Its length is 8 and it is longer than the unitary aliquot sequences of all the numbers below 134.

References

  • Richard K. Guy, "Unitary aliquot sequences", Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. B8, pp. 97-99.
  • Richard K. Guy and Marvin C. Wunderlich, Computing Unitary Aliquot Sequences: A Preliminary Report, University of Calgary, Department of Mathematics and Statistics, 1979.
  • H. J. J. te Riele, Unitary Aliquot Sequences, MR 139/72, Mathematisch Centrum, 1972, Amsterdam.
  • H. J. J. te Riele, Further Results On Unitary Aliquot Sequences. NW 2/73, Mathematisch Centrum, 1973, Amsterdam.

Links

Crossrefs

Cf. A098008, A098009, A098010, A127652, A127653, A127654, A127655, A290144.

Programs

  • Mathematica
    usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])];
g[n_] := If[n > 0, usigma[n] - n, 0]; f[n_] := NestWhileList[g, n, UnsameQ, All]; a = -1; seq = {}; Do[b = Length[f[n]] - 1; If[b > a, a = b; AppendTo[seq, n]], {n, 10^6}] ; seq (* after Giovanni Resta at A034448 & Robert G. Wilson v at A098009 *)

A290144 Record lengths of transient part of the unitary aliquot sequences of the numbers in A290143.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 43, 45, 67, 78, 205, 207, 1109, 1116, 1117, 1155, 1162, 1163, 1171, 1711, 1712, 1828, 1829
Offset: 1

Views

Author

Amiram Eldar, Jul 21 2017

Keywords

Examples

			The unitary aliquot sequence of 134 is: 134, 70, 74, 40, 14, 10, 8, 1. Its length is 8 and it is longer than the unitary aliquot sequences of all the numbers below 134.

Crossrefs

Cf. A098008, A098009, A098010, A127652, A127653, A127654, A127655, A290143.

Programs

  • Mathematica
    usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])];
g[n_] := If[n > 0, usigma[n] - n, 0]; f[n_] := NestWhileList[g, n, UnsameQ, All]; a = 0; seq = {}; Do[b = Length[f[n]] - 2; If[b > a, a = b; AppendTo[seq, b]], {n, 10^6}]; seq (* after Giovanni Resta at A034448 & Robert G. Wilson v at A098009 *)
Showing 1-5 of 5 results.

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