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 1-10 of 13 results. Next

A063050 a(n) = number of 'Reverse and Add!' operations that have to be applied to the n-th term of A063049 in order to obtain a term in the trajectory of 196.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 0, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1
Offset: 0

Views

Author

Klaus Brockhaus, Jul 07 2001

Keywords

Examples

			394 is a term of A063049. One 'Reverse and Add!' operation applied to 394 leads to a term (887) in the trajectory of 196, so the corresponding term of the present sequence is 1.

Links

Crossrefs

A023108, A033865, A006960, A063049.

Programs

  • Mathematica
    limit = 10^3; x = NestList[ # + IntegerReverse[#] &, 196, limit];
y = Select[Range[197, 4942],
   Intersection[NestList[ # + IntegerReverse[#] &, #, limit],
      x] != {} &];
Table[
 Length@NestWhileList[# + IntegerReverse[#] &,
    y[[i]], ! MemberQ[x, #] &] - 1, {i, Length[y]}]
(* Robert Price, Oct 21 2019 *)

A063048 Numbers n such that the Reverse and Add! trajectory of n (presumably) does not reach a palindrome and does not join the trajectory of any term m < n.

Original entry on oeis.org

196, 879, 1997, 7059, 10553, 10563, 10577, 10583, 10585, 10638, 10663, 10668, 10697, 10715, 10728, 10735, 10746, 10748, 10783, 10785, 10787, 10788, 10877, 10883, 10963, 10965, 10969, 10977, 10983, 10985, 12797, 12898, 13097, 13197, 13694
Offset: 1

Views

Author

Klaus Brockhaus, Jul 07 2001, revised Nov 04 2003

Keywords

Comments

The starting number n is regarded as part of the trajectory, so palindromes are excluded from the sequence. A088753 is obtained if palindromes are not excluded. The smallest term in A063048 but not in A088753 is 19098, the smallest term in A088753 but not in A063048 is 9999.
Subsequence of A023108. Sequence A070788 is similarly defined, but palindromes are irrelevant. Corresponding sequences for other bases are A075252 (base 2), A077405 (base 3), A075421 (base 4).
If the trajectory of a number k joins the trajectory of a smaller number which is a term of the present sequence, then this occurs after very few Reverse and Add! steps (at most 8 for k < 100000, at most 10 for k < 1000000). On the other hand, the trajectories of the terms < 14000 do not join the trajectory of any smaller term within at least 1500 steps. This is the precise meaning of "presumably" in the definition.
The terms are rather unevenly distributed. They form clusters, especially above 10^4, 10^5, 10^6, ... . The interval from 10000 to 11000 for example contains 26 terms, whereas only two terms occur in the interval from 90000 to 100000.
It seems that if the most significant digit is not equal to 1, the least significant digit is always 9, while this does not hold for the Lychrel numbers as in A023108. - A.H.M. Smeets, Feb 18 2019
From A.H.M. Smeets, Sep 18 2021: (Start)
Let d_0 d_1 d_2 ... d_n be the decimal digits of an (n+1)-digit number.
All numbers in this sequence seem to satisfy the following condition:
d_0 = "1" or d_n = "9", and for all k, 0 < k < floor((n-1)/2), d_k = "0" or d_k = "9" or d_(n-k) = "0" or d_(n-k) = "9".
The plot log_10(a(n)) versus log_10(n) shows a stepwise behavior. However, the global behavior seems to be a straight line with slope e/(e-1) (= A185393). This slope is also obtained for the seeds in the Reverse and Add! problem in other bases. (End)

Examples

			1997 is a term since the trajectory of 1997 (presumably) does not lead to a number which occurs in the trajectory of 196 or of 879 (actually checked for the first 10000 terms of these trajectories). The trajectory of 1997 joins the trajectory of 106 at 97768 (cf. A070796), but 106 is not a term of the present sequence.

References

  • Daniel Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, Paris, 2012, 702 pages. See Entry 196.

Links

Crossrefs

Cf. A033865, A023108, A006960, A063049 - A063065, A088753, A070788, A075252, A077405, A075421, A070796.

Programs

  • Mathematica
    limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
utraj = {};
Select[Range[0, 14000], (x = NestWhileList[ # + IntegerReverse[#] &, #, ! PalindromeQ[#] &, 1, limit];
   If[Length[x] >= limit && Intersection[x, utraj] == {},
    utraj = Union[utraj, x]; True,
utraj = Union[utraj, x]]) &] (* Robert Price, Oct 16 2019 *)

A070788 Positive integers n such that the Reverse and Add! trajectory of n (presumably) does not join the trajectory of any m < n.

Original entry on oeis.org

1, 3, 5, 7, 9, 100, 102, 106, 108, 111, 112, 113, 114, 116, 117, 118, 119, 122, 124, 128, 133, 135, 137, 138, 166, 184, 186, 196, 199, 359, 399, 459, 539, 659, 679, 739, 759, 779, 799, 859, 879, 919, 939, 959, 979, 999, 1000, 1006, 1011, 1013, 1022, 1033, 1037
Offset: 1

Views

Author

Klaus Brockhaus, May 07 2002, revised Oct 15 2003

Keywords

Comments

The conjecture that the trajectories of the terms of this sequence do not join is based on the observation that if the trajectories of two integers below 10000 join, this happens after at most 15 steps, while for any two terms the trajectories do not join within 1200 steps. For pairs from 1, 3, 5, 7, 9, 100, 102, 106 this has even been checked for 10000 steps.
The positive integers are the domain of the equivalence relation 'the trajectories of a and b join'; each of its presumably infinitely many equivalence classes is represented by a term of this sequence. Each class contains infinitely many integers (cf. A070789 - A070798). In such a class the relation 'the trajectory of a is part of the trajectory of b' is a partial order for which a term c is a maximal element if it is in A067031 (integers not of the form k + reverse(k) for any k) and the integer at which the trajectories of a and b join is the greatest lower bound of a and b.

Examples

			The trajectory of 2 is part of the trajectory of 1; the trajectory of 3 does not join the trajectory of 1 within 10000 steps; the trajectory of 5 does not join the trajectory of 1 or of 3 within 10000 steps.

Links

Crossrefs

Cf. A006960, A063048, A067031, A070789 - A070798, A063049.

Programs

  • Mathematica
    limit = 10^3; utraj = {};
Select[Range[1037], (x = NestList[ # + IntegerReverse[#] &, #, limit]; If[Intersection[x, utraj] == {}, utraj = Union[utraj, x]; True, utraj = Union[utraj, x]]) &] (* Robert Price, Oct 20 2019 *)

A070798 Integers n such that the 'Reverse and Add!' trajectory of n joins the trajectory of 111.

Original entry on oeis.org

111, 123, 147, 210, 222, 246, 321, 345, 420, 444, 543, 642, 741, 789, 840, 888, 987, 1002, 1005, 1596, 1686, 1776, 1866, 1956, 2001, 2004, 2595, 2685, 2775, 2865, 2955, 3000, 3003, 3009, 3099, 3189, 3279, 3369, 3459, 3549, 3594, 3639, 3684, 3729, 3774
Offset: 1

Views

Author

Klaus Brockhaus, May 07 2002

Keywords

Comments

It appears that the first differences of this sequence are always a multiple of 3. - Robert Price, Oct 20 2019

Examples

			The trajectory of 222 is part of the trajectory of 111; the trajectory of 2004 joins the trajectory of 111 at 66066 after 7 steps.

Links

Crossrefs

Cf. A070788 - A070797, A063049.

Programs

  • Mathematica
    limit = 10^2; x = NestList[ # + IntegerReverse[#] &, 111, limit];
Select[Range[3774],
 Intersection[NestList[ # + IntegerReverse[#] &, #, limit],
x] != {} &] (* Robert Price, Oct 20 2019 *)

A070791 Integers n such that the 'Reverse and Add!' trajectory of n joins the trajectory of 5.

Original entry on oeis.org

5, 10, 11, 13, 17, 20, 22, 26, 31, 35, 40, 44, 53, 62, 71, 79, 80, 88, 97, 115, 142, 158, 170, 176, 185, 214, 241, 257, 275, 284, 313, 329, 340, 356, 374, 383, 412, 428, 455, 469, 473, 482, 511, 527, 554, 568, 572, 581, 599, 610, 626, 649, 653, 667, 671, 680
Offset: 1

Views

Author

Klaus Brockhaus, May 07 2002

Keywords

Examples

			The trajectory of 10 is part of the trajectory of 5; the trajectory of 340 joins the trajectory of 5 at 88088 after 7 steps.

Links

Crossrefs

Cf. A070788 - A070790, A070792 - A070798, A063049.

Programs

  • Mathematica
    limit = 10^3; x = NestList[ # + IntegerReverse[#] &, 5, limit];
Select[Range[680],
 Intersection[NestList[ # + IntegerReverse[#] &, #, limit],
x] != {} &] (* Robert Price, Oct 20 2019 *)

A070796 Integers n such that the 'Reverse and Add!' trajectory of n joins the trajectory of 106.

Original entry on oeis.org

106, 152, 179, 205, 251, 278, 304, 350, 377, 379, 403, 476, 478, 479, 502, 509, 575, 577, 578, 601, 608, 674, 676, 677, 700, 707, 773, 775, 776, 806, 872, 874, 875, 905, 971, 973, 974, 1019, 1054, 1060, 1082, 1093, 1109, 1144, 1150, 1172, 1183, 1199
Offset: 1

Views

Author

Klaus Brockhaus, May 07 2002

Keywords

Examples

			The trajectory of 707 is part of the trajectory of 106; the trajectory of 775 joins the trajectory of 106 at 48884 after 5 steps.

Links

Crossrefs

Cf. A070788 - A070795, A070797, A070798, A063049.

Programs

  • Mathematica
    limit = 10^3; x = NestList[ # + IntegerReverse[#] &, 106, limit];
Select[Range[1199],
 Intersection[NestList[ # + IntegerReverse[#] &, #, limit],
x] != {} &] (* Robert Price, Oct 20 2019 *)

A070789 Integers n such that the 'Reverse and Add!' trajectory of n joins the trajectory of 1.

Original entry on oeis.org

1, 2, 4, 8, 16, 25, 34, 43, 52, 59, 61, 68, 70, 77, 86, 95, 104, 109, 151, 154, 155, 157, 203, 208, 209, 250, 253, 254, 256, 302, 307, 308, 352, 353, 355, 401, 406, 407, 409, 451, 452, 454, 500, 505, 506, 508, 550, 551, 553, 604, 605, 607, 650, 652, 703, 704
Offset: 1

Views

Author

Klaus Brockhaus, May 07 2002

Keywords

Examples

			The trajectory of 2 is part of the trajectory of 1; the trajectory of 401 joins the trajectory of 1 at 1111 after 3 steps.

Links

Crossrefs

Cf. A070788, A070790 - A070798, A063049.

Programs

  • Mathematica
    limit = 10^3; x = NestList[ # + IntegerReverse[#] &, 1, limit];
Select[Range[704],
 Intersection[NestList[ # + IntegerReverse[#] &, #, limit],
x] != {} &] (* Robert Price, Oct 20 2019 *)

A070790 Integers n such that the 'Reverse and Add!' trajectory of n joins the trajectory of 3.

Original entry on oeis.org

3, 6, 12, 15, 21, 24, 30, 33, 39, 42, 48, 51, 57, 60, 66, 69, 75, 78, 84, 87, 93, 96, 132, 159, 165, 177, 219, 231, 258, 264, 276, 318, 330, 357, 363, 375, 417, 429, 456, 462, 474, 516, 528, 555, 561, 573, 579, 615, 627, 654, 660, 672, 678, 699, 714, 726, 753
Offset: 1

Views

Author

Klaus Brockhaus, May 07 2002

Keywords

Examples

			The trajectory of 6 is part of the trajectory of 3; the trajectory of 375 joins the trajectory of 3 at 9768 after 3 steps.

Links

Crossrefs

Cf. A070788, A070789, A070791 - A070798, A063049.

Programs

  • Mathematica
    limit = 10^3; x = NestList[ # + IntegerReverse[#] &, 3, limit];
Select[Range[753],
 Intersection[NestList[ # + IntegerReverse[#] &, #, limit],
x] != {} &] (* Robert Price, Oct 20 2019 *)

A070792 Integers n such that the 'Reverse and Add!' trajectory of n joins the trajectory of 7.

Original entry on oeis.org

7, 14, 19, 23, 28, 29, 32, 37, 38, 41, 46, 47, 49, 50, 55, 56, 58, 64, 65, 67, 73, 74, 76, 82, 83, 85, 89, 91, 92, 94, 98, 110, 121, 136, 143, 187, 220, 235, 242, 286, 334, 341, 385, 433, 440, 484, 532, 569, 583, 631, 668, 682, 719, 730, 767, 781, 818, 866, 869
Offset: 1

Views

Author

Klaus Brockhaus, May 07 2002

Keywords

Examples

			The trajectory of 14 is part of the trajectory of 7; the trajectory of 235 joins the trajectory of 7 at 8872688 after 13 steps.

Links

Crossrefs

Cf. A070788 - A070791, A070793 - A070798, A063049.

Programs

  • Mathematica
    limit = 10^3; x = NestList[ # + IntegerReverse[#] &, 7, limit];
Select[Range[869],
 Intersection[NestList[ # + IntegerReverse[#] &, #, limit],
x] != {} &] (* Robert Price, Oct 20 2019 *)

A070793 Integers n such that the 'Reverse and Add!' trajectory of n joins the trajectory of 9.

Original entry on oeis.org

9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 198, 297, 396, 495, 594, 693, 792, 891, 990, 1017, 1035, 1044, 1089, 1107, 1125, 1134, 1179, 1215, 1224, 1269, 1305, 1314, 1359, 1404, 1449, 1539, 1629, 1719, 1809, 1899, 1989, 2016, 2034, 2043, 2088, 2106, 2124
Offset: 1

Views

Author

Klaus Brockhaus, May 07 2002

Keywords

Examples

			The trajectory of 18 is part of the trajectory of 9; the trajectory of 1215 joins the trajectory of 9 at 40293 after 3 steps.

Links

Crossrefs

Cf. A070788 - A070792, A070794 - A070798, A063049.

Programs

  • Mathematica
    limit = 10^3; x = NestList[ # + IntegerReverse[#] &, 9, limit];
Select[Range[2124],
 Intersection[NestList[ # + IntegerReverse[#] &, #, limit],
x] != {} &] (* Robert Price, Oct 20 2019 *)
Showing 1-10 of 13 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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