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.

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A090071 Numbers n such that there are (presumably) ten palindromes in the Reverse and Add! trajectory of n.

Original entry on oeis.org

2, 5, 10003, 30001, 40000, 40004, 100000, 100001, 2000000, 2000002
Offset: 1

Views

Author

Klaus Brockhaus, Nov 20 2003

Keywords

Comments

Additional terms are 20000000, 20000002, 200000000, 200000002, 2000000000, 2000000002, 10000000004, 10000100001, 20000000000, 20000000002, 20000000003, 30000000002, 40000000001, but it is not yet ascertained that they are consecutive.
For all terms given above each palindrome is reached from the preceding one or from the start in at most 35 steps; after the presumably last one no further palindrome is reached in 5000 steps.

Examples

			The trajectory of 2 begins 2, 4, 8, 16, 77, 154, 605, 1111, 2222, 4444, 8888, 17776, 85547, 160105, 661166, 1322332, 3654563, 7309126, ...; at 7309126 it joins the (presumably) palindrome-free trajectory of A063048(7) = 10577, hence 2, 4, 8, 77, 1111, 2222, 4444, 8888, 661166 and 3654563 are the ten palindromes in the trajectory of 2 and 2 is a term.

Links

Crossrefs

Cf. A023108, A023109, A065001, A070742, A077594.

A090072 Numbers n such that there are (presumably) eleven palindromes in the Reverse and Add! trajectory of n.

Original entry on oeis.org

1, 20000, 20002, 1000000, 1000001, 10000000, 10000001
Offset: 1

Views

Author

Klaus Brockhaus, Nov 20 2003

Keywords

Comments

Additional terms (cf. A090075) are 100000000, 100000001, 100010001, 1000000000, 1000000001, 10000000000, 10000000001, 100000000000, 100000000001, 1000000000000, 1000000000001, 1000001000001, 1000100010001, but it is not yet ascertained that they are consecutive.
For all terms given above each palindrome is reached from the preceding one or from the start in at most 35 steps; after the presumably last one no further palindrome is reached in 5000 steps.
Only two numbers are known whose Reverse and Add trajectory contains twelve palindromes: 10000 and 10001. It is conjectured that these are the only such numbers and it has been conjectured before (cf. A077594) that no Reverse and Add trajectory contains more than twelve palindromes.

Examples

			The trajectory of 1 begins 1, 2, 4, 8, 16, 77, 154, 605, 1111, 2222, 4444, 8888, 17776, 85547, 160105, 661166, 1322332, 3654563, 7309126, ...; at 7309126 it joins the (presumably) palindrome-free trajectory of A063048(7) = 10577, hence 1, 2, 4, 8, 77, 1111, 2222, 4444, 8888, 661166 and 3654563 are the eleven palindromes in the trajectory of 1 and 1 is a term.

Links

Crossrefs

Cf. A023108, A023109, A065001, A070742, A077594, A090075.

A286481 Numbers which require exactly 260 'Reverse and Add' steps to reach a palindrome.

Original entry on oeis.org

1003062289999939142, 1003062299899939142, 1003062389989939142, 1003062399889939142, 1003062489979939142, 1003062499879939142, 1003062589969939142, 1003062599869939142, 1003062689959939142, 1003062699859939142, 1003062789949939142, 1003062799849939142, 1003062889939939142, 1003062899839939142, 1003062989929939142, 1003062999829939142
Offset: 1

Views

Author

Andrey S. Shchebetov and Sergei D. Shchebetov, May 12 2017

Keywords

Comments

The sequence starts with 1003062289999939142 (the 19-digit number discovered by Vaughn Suite on Mar 19 2006) and continues for another 430079 terms (none previously reported) each turning into a 119-digit palindrome after 260 steps until the sequence ends with 3419399999822603000 (see a-file). No further numbers beyond 3419399999822603000 belonging to the same sequence exist. The sequence was predicted theoretically and found empirically using computer modeling algorithms. For the first 100 terms of the sequence see b-file.

Examples

			a(1) = 1003062289999939142 + 2419399999822603001 = 3422462289822542143

References

  • Popular Computing (Calabasas, CA), The 196 Problem, Vol. 3 (No. 30, Sep 1975).

Links

Crossrefs

Cf. A023109, A033672, A065198, A065199, A065320, A065321, A065322, A065323, A065324, A065325, A065326, A065327, A070743, A072216, A072217, A072218, A281301, A281390, A281506, A281507, A281508, A281509.

Formula

a(n+1) = a(n) + rev(a(n)).

A033866 Least number of Reverse-then-add persistence n.

Original entry on oeis.org

1, 10, 19, 59, 69, 166, 79, 188, 193, 1397, 829, 167, 2069, 1797, 849, 177, 1496, 739, 1798, 10777, 6999, 1297, 869, 187, 89, 10797, 10853, 10921, 10971, 13297, 10548, 13293, 17793, 20889, 700269, 106977, 108933, 80359, 13697, 10794, 15891
Offset: 0

Views

Author

David W. Wilson

Keywords

Crossrefs

Essentially same as A023109.

A077403 In base 3: smallest number that requires n Reverse and Add! steps to reach a palindrome.

Original entry on oeis.org

0, 3, 5, 15, 17, 263, 170, 509, 491, 322, 266, 222, 161, 494, 260, 106, 95, 78, 53, 2425, 1466, 9717, 59583, 38878, 38798, 33515, 39440, 32857, 37340, 238849, 177470, 60019, 59655, 178540, 124895, 59753, 179751, 1595576, 715615, 354605, 179575
Offset: 0

Views

Author

Klaus Brockhaus, Nov 05 2002

Keywords

Comments

Base 3 analog of A066058 (base 2), A077441 (base 4) and A023109 (base 10).

Examples

			5 is the smallest number which requires two steps to reach a base 3 palindrome (cf. A066057), so a(2) = 5; 5 (decimal) = 12 -> 12 + 21 = 110 -> 110 + 011 = 121 (palindrome) = 16 (decimal).

Links

Crossrefs

Cf. A066057, A066058, A077441, A023109.

Programs

  • ARIBAS
    var ar: array; end; m := 45; ar := alloc(array, m+1, -1); mc := m+1; n := 0; while mc > 0 do v := -1; c := 0; k := n; while c < m+1 do d := k; rev := 0; while d > 0 do rev := 3*rev+(d mod 3); d := d div 3; end; if k = rev then v := c; c := m+1; else inc(c); k := k+rev; end; end; if 0 <= v and v <= m then if ar[v] < 0 then ar[v] := n; dec(mc); write((v,n)); end; end; inc(n); end; writeln(); for j := 0 to m do write(ar[j],","); end;

A243824 Two-column array A(n,s) of pairs (n,s) read by row where s is the smallest seed number such that the Reverse and Add! trajectory of s contains n (excluding cases where n=s).

Original entry on oeis.org

2, 1, 4, 1, 6, 3, 8, 1, 10, 5, 11, 5, 12, 3, 14, 7, 16, 1, 18, 9, 22, 5
Offset: 2

Views

Author

Felix Fröhlich, Jun 11 2014

Keywords

Examples

			A(10,1)=16 is in the array because 16 is the 9th number appearing in the Reverse and Add! trajectory of a smaller number.
A(10,2)=1 is in the array because 1 + 1 = 2, 2 + 2 = 4, 4 + 4 = 8, 8 + 8 = 16, so 1 is the smallest seed number whose Reverse and Add! trajectory contains 16.
Array begins:
  2 1
  4 1
  6 3
  8 1
  10 5
  11 5
  12 3
  14 7
  16 1
  18 9
  22 5

Links

Crossrefs

Cf. A006960, A023109, A033665, A063048, A070788, A077594.

A353185 Numbers which require exactly 289 'Reverse and Add' steps to reach a palindrome.

Original entry on oeis.org

10037000230509917799950, 10037000240508917799950, 10037000250507917799950, 10037000260506917799950, 10037000270505917799950, 10037000280504917799950, 10037000290503917799950, 10037000330509817799950, 10037000340508817799950
Offset: 1

Views

Author

Andrey S. Shchebetov and Sergei D. Shchebetov, Apr 29 2022

Keywords

Comments

The sequence starts with 10037000230509917799950, ends with 15999771990503200073000 and contains 9031680 terms known at present, including 13968441660506503386020 and 13568441660506503386420 discovered by Anton Stefanov on January 5, 2021.

References

  • Popular Computing (Calabasas, CA), The 196 Problem, Vol. 3 (No. 30, Sep 1975).

Links

Crossrefs

A023109, A033672, A065198, A065199, A065320, A065321, A065322, A065323, A065324, A065325, A065326, A065327, A070743, A072216, A072217, A072218, A281301, A281390, A281506, A281507, A326414.

Formula

Each term requires exactly 289 steps to turn into a 142-digit palindrome.

A090062 There is (presumably) one and only one palindrome in the Reverse and Add! trajectory of n.

Original entry on oeis.org

89, 98, 167, 187, 266, 286, 365, 385, 479, 563, 578, 583, 662, 677, 682, 749, 761, 776, 779, 781, 829, 860, 869, 875, 880, 899, 928, 947, 968, 974, 977, 998, 1077, 1093, 1098, 1167, 1183, 1188, 1257, 1273, 1278, 1297, 1347, 1363, 1368, 1387, 1396, 1397, 1437
Offset: 1

Views

Author

Klaus Brockhaus, Nov 20 2003

Keywords

Comments

For terms < 2000 the only palindrome is reached from the start in at most 24 steps; thereafter no further palindrome is reached in 2000 steps.

Examples

			The trajectory of 479 begins 479, 1453, 4994, 9988, 18887, ...; at 9988 it joins the (presumably) palindrome-free trajectory of A063048(3) = 1997, hence 4994 is the only palindrome in the trajectory of 479 and 479 is a term.

Links

Crossrefs

Cf. A023108, A023109, A065001, A070742, A077594.

A090063 Numbers n such that there are (presumably) two palindromes in the Reverse and Add! trajectory of n.

Original entry on oeis.org

49, 58, 67, 76, 85, 94, 108, 118, 127, 133, 143, 148, 153, 173, 177, 178, 198, 207, 217, 226, 239, 247, 276, 277, 279, 297, 306, 316, 325, 331, 338, 339, 341, 346, 349, 351, 371, 375, 376, 378, 379, 396, 405, 415, 419, 430, 437, 438, 440, 445, 448, 450, 464
Offset: 1

Views

Author

Klaus Brockhaus, Nov 20 2003

Keywords

Comments

For terms < 2000 each palindrome is reached from the preceding one or from the start in at most 24 steps; after the presumably last one no further palindrome is reached in 2000 steps.

Examples

			The trajectory of 118 begins 118, 929, 1858, 10439, 103840, 152141, 293392, 586784, 1074469, ...; at 1074469 it joins the (presumably) palindrome-free trajectory of A063048(72) = 90379, hence 929 and 293392 are the two palindromes in the trajectory of 118 and 118 is a term.

Links

Crossrefs

Cf. A023108, A023109, A065001, A070742, A077594.

A090064 Numbers n such that there are (presumably) three palindromes in the Reverse and Add! trajectory of n.

Original entry on oeis.org

18, 27, 36, 45, 54, 63, 69, 72, 78, 81, 87, 90, 96, 99, 113, 125, 126, 128, 137, 146, 149, 156, 157, 162, 163, 165, 168, 169, 172, 175, 180, 183, 188, 189, 193, 194, 195, 197, 220, 224, 225, 227, 232, 236, 242, 245, 248, 252, 255, 256, 259, 261, 264, 267, 268
Offset: 1

Views

Author

Klaus Brockhaus, Nov 20 2003

Keywords

Comments

For terms < 2000 each palindrome is reached from the preceding one or from the start in at most 24 steps; after the presumably last one no further palindrome is reached in 2000 steps.

Examples

			The trajectory of 113 begins 113, 424, 848, 1696, 8657, 16225, 68486, 136972, 416603, ...; at 416603 it joins the (presumably) palindrome-free trajectory of A063048(16) = 10735, hence 424, 848 and 68486 are the three palindromes in the trajectory of 113 and 113 is a term.

Links

Crossrefs

Cf. A023108, A023109, A065001, A070742, A077594.
Previous Showing 21-30 of 34 results. Next

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