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 1-4 of 4 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

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 1-4 of 4 results.

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