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 15 results. Next

A171740 The first number palindromic with n digits in more than one base (shown in base 10).

Original entry on oeis.org

1, 8, 26, 624, 2293, 207702, 186621, 342324801, 27924649, 260311602096, 1556085529, 248876637484140, 318713056300, 2544221971606336, 4712469842177, 530386561769238496, 1939137135947326
Offset: 1

Views

Author

James G. Merickel, Dec 17 2009

Keywords

Comments

This sequence gives the first value (represented in base 10) that is palindromic -- i.e., the same when its digits are reflected about the center -- with n >= 1 digits in two different bases.
Currently waiting on doubly 18-palindrome. The 19-palindrome is one in bases 10 and 11, 6411682614162861146 in base 10. For some time I was laboring under the impression that the new terms now added -- starting at 9-palindromes -- were too difficult to find, thinking a change I recently made would not increase programming efficiency as much as it did. The results come from the research done for the sequences labeled with A216*** shown below. The other cross-referenced sequences, A171***, deal with higher orders of multiplicity and such things. - James G. Merickel, Sep 19 2012

Examples

			a(2)=8 is 11 in base 7 and 22 in base 3.
a(3)=26 is 101 in base 5 and 222 in base 3.
a(4)=624 is 1551 in base 7 and 4444 in base 5.
a(7)=186621 is 3555553 in base 6 and 1405041 in base 7.
a(8)=342324801 is 96788769 in base 12 and 14677641 in base 16.

Crossrefs

Cf. A171701, A171702, A171703, A171704, A171705, A171706, A171741, A171742, A216840, A216841, A216843, A216899, A216900, A216901, A216902, A216903, A216904, A216905, A216906, A216907, A216908, A216909, A216910.

Extensions

Corrected typo in example by Chai Wah Wu, Jul 18 2015

A216840 Smallest palindromic number of 3 digits in two bases differing by n.

Original entry on oeis.org

46, 26, 121, 109, 157, 211, 209, 197, 257, 307, 381, 576, 463, 676, 601, 701, 757, 842, 929, 1086, 1123, 1445, 1333, 1717, 1297, 1801, 1522, 2092, 1765, 2393, 2026, 2026, 2305, 2696, 2501, 2701, 2757, 2971, 3133, 3600, 3421, 3718, 4411, 3845, 4551, 4031, 4691
Offset: 1

Views

Author

James G. Merickel, Sep 19 2012

Keywords

Comments

This is one of a collection of sequences of doubly palindromic numbers of same lengths in each of two bases. The lengths go from 2 through 17, excluding 16 (only one term available at present), and the order of the first two of these are switched in terms of their A-numbers. These are the A216*** cross-references. The 171*** c-refs are to a variety of record multiple-base palindromes. Larger comments are to be found -- will generally be -- in the 2-palindrome sequence. The smaller bases of a pair here are (in sequence) 4, 3, 6, 5, 6, 7, 6, 6, 7, 7.

Examples

			The first entry here, 46 in base 10, is represented as 222 in base 4 and 141 in base 5. The 2nd entry here, 26 in base 10, is represented as 222 in base 3 and 101 in base 5. The next is then the smallest in bases that differ by 3, bases 6 and 9 by what is in the comment.
a(3) = 121 is 232 in base 7, a(5) = 157 is 313 in base 7 and 111 in base 12, a(6) = 211 is 323 in base 8 and 111 in base 14. - _Chai Wah Wu_, Aug 19 2015

Links

Crossrefs

Cf. A171701, A171702, A171703, A171704, A171705, A171706, A171740, A171741, A171742, A171775, A216841, A216843, A216899, A216900, A216901, A216902, A216903, A216904, A216905, A216906, A216907, A216908, A216909, A216910.

Extensions

More terms and corrected a(3), a(5) and a(6) by Chai Wah Wu, Aug 19 2015

A216841 Smallest palindromic number of length 2 in two bases differing by n.

Original entry on oeis.org

24, 18, 8, 10, 12, 14, 16, 18, 15, 22, 18, 26, 21, 30, 24, 34, 24, 38, 30, 28, 33, 46, 32, 50, 39, 36, 35, 58, 40, 62, 40, 44, 51, 70, 45, 74, 57, 52, 48, 82, 56, 86, 55, 54, 69, 94, 60, 98, 60, 68, 65, 106, 63, 66, 70, 76, 87, 118, 70, 122, 93, 84, 80, 78, 77
Offset: 2

Views

Author

James G. Merickel, Sep 19 2012

Keywords

Comments

This and other sequences in this collection -- that runs through 17-digit palindromes but (for now) excludes 16-digit ones (but see A216910) -- have offset 2 because an even-length palindrome in one base ends in 0 in the base one larger. After its first two terms, this particular sequence in the collection is trivial. The collection in its entirety are the A216*** cross-references plus this one. The smaller of the pair of bases here are (in sequence) 5, 5, 3, 4, 5, 6, 7, 8, 9. Aside from the first two sequences being switched in order of their A-numbers, the others are in order (but note that only the last 12 are without gaps in the A-sequencing). The A171*** cross-references are to a variety of record small multi-base palindromes.

Examples

			All numbers smaller than 24 (in base 10) fail to have two bases differing by 2 in which the number is a 2-digit palindrome.  Decimal number 24 is 44 in base 5 and is 33 in base 7.  Similarly, for the second term here, the facts that decimal number 18 is 33 in base 5 and 22 in base 8 and that no smaller number than decimal 18 has 2 bases in which it is a 2-digit palindrome result in its position here.
a(10) = 15 is 33 in base 4 and 11 in base 14. - _Chai Wah Wu_, Aug 19 2015

Links

Crossrefs

Cf. A171701, A171702, A171703, A171704, A171705, A171706, A171740, A171741, A171742, A171775, A216840, A216843, A216899, A216900, A216901, A216902, A216903, A216904, A216905, A216906, A216907, A216908, A216909, A216910.

Extensions

More terms and corrected a(10) from Chai Wah Wu, Aug 19 2015

A216843 Smallest palindromic numbers of length 4 in two bases differing by n.

Original entry on oeis.org

624, 910, 19040, 17520, 7056, 16842, 13320, 42876, 30480, 27650, 14376, 18226, 22708, 27870, 33760, 50524, 71856, 70338, 55981, 113694, 107888, 75810, 139920, 223000, 140608, 157464, 175616, 195112, 125000, 238328, 262144, 287496, 314432, 343000, 238329
Offset: 2

Views

Author

James G. Merickel, Sep 19 2012

Keywords

Comments

See A216841 and A216840 for their comment sections. The A171*** c-refs are to various record small multi-base palindromes. The others are the remainder of this collection.

Examples

			624 (in decimal), the 1st term here, is 4444 in base 5 and 1551 in base 7.

Links

Crossrefs

Cf. A171701, A171702, A171703, A171704, A171705, A171706, A171740, A171741, A171742, A171775, A216840, A216841, A216899, A216900, A216901, A216902, A216903, A216904, A216905, A216906, A216907, A216908, A216909, A216910.

Extensions

More terms from Chai Wah Wu, Aug 16 2015

A216899 Smallest palindromic number of 5 digits in two bases differing by n.

Original entry on oeis.org

2293, 5001, 15731, 23341, 31279, 90209, 94249, 131766, 221866, 380809, 589816, 499617, 661753, 1219745, 999503, 1456663, 1722526, 2376918, 2552395, 2450410, 2983858, 4317391, 5582689, 4490153, 4934907, 6298020, 7410454, 8531146, 9232273, 11891740, 11309998
Offset: 1

Views

Author

James G. Merickel, Sep 19 2012

Keywords

Comments

Part of a collection. See comments sections of A216840 and A216841. Smaller of base pairs are 5, 6, 7, 8, 8, 10, 10, 11, 12, 14. The A171*** c-refs are to various record multi-base palindromes.

Examples

			2293 is 33133 in base 5 and is 14341 in base 6.  No smaller number is a 5-digit palindrome in both of two bases differing by 1.
15731 is 35653 in base 8 and 10901 in base 11. - _Chai Wah Wu_, Aug 20 2015

Links

Crossrefs

Cf. A171701, A171702, A171703, A171704, A171705, A171706, A171740, A171741, A171742, A171775, A216840, A216841, A216843, A216900, A216901, A216902, A216903, A216904, A216905, A216906, A216907, A216908, A216909, A216910.

Extensions

More terms and corrected a(3) and a(10) by Chai Wah Wu, Aug 20 2015

A216900 Smallest palindromic number of length 6 in two bases differing by n.

Original entry on oeis.org

207702, 1102816, 546645, 34028500, 1487484, 10652864, 17438409, 55257336, 12338379, 39957792, 81828216, 235790724, 120261375, 75954970, 166571505, 733418352, 276835725, 784438752, 169796925, 433366956, 1390393356, 2502432704, 984536776, 5008282776, 2289422850
Offset: 2

Views

Author

James G. Merickel, Sep 19 2012

Keywords

Comments

Part of a collection. See comments in A216840 and A216841. Smaller bases begin with 8, 11, 10, 19, 13. The A171*** c-refs are to various record multi-base palindromes. The A216*** ones are the others in this collection.

Examples

			207702 is also a palindrome of 6 digits in base 8, and no smaller number is a palindrome of 6 digits in both of two bases.
1102816 in base 12 and 15 are 452254 and 16bb61 respectively. - _Chai Wah Wu_, Aug 19 2015

Links

Crossrefs

Cf. A171701, A171702, A171703, A171704, A171705, A171706, A171740, A171741, A171742, A171775, A216840, A216841, A216843, A216899, A216901, A216902, A216903, A216904, A216905, A216906, A216907, A216908, A216909, A216910.

Extensions

More terms and corrected a(3) from Chai Wah Wu, Aug 19 2015

A216901 Smallest palindromic number of 7 digits in two bases differing by n.

Original entry on oeis.org

186621, 532900, 1857351, 9046409, 17439233, 69012421, 146766357, 289744476, 347411351, 390219121, 1200809057, 1393838392, 3088676989, 2383346701, 3333147809, 6041072349, 6819663132, 7822398453, 9534770061, 12359264257, 18701296795, 20196246357, 25763014657
Offset: 1

Views

Author

James G. Merickel, Sep 19 2012

Keywords

Comments

See comments to A216840 and A216841. Smaller of pair of bases are 6, 7, 8, 10, 11, 14, 15, 17, 17, 17. The A216*** cross-references here are other sequences of this collection, while the A171*** ones are to various multi-base record palindromes.

Examples

			9046409 is also a 7-digit palindrome in base 14 (in addition to decimal).  No smaller number is a 7-digit palindrome in both of two bases that differ by 4.

Links

Crossrefs

Cf. A171701, A171702, A171703, A171704, A171705, A171706, A171740, A171741, A171742, A171775, A216840, A216841, A216843, A216899, A216900, A216902, A216903, A216904, A216905, A216906, A216907, A216908, A216909, A216910.

Extensions

More terms from Chai Wah Wu, Aug 16 2015

A216905 Smallest palindromic number of 11 digits in two bases differing by n.

Original entry on oeis.org

1556085529, 28455496928, 671667243106, 6286726505905, 17441819065567, 96396891300626, 470389217692631, 899379244450606, 2169311631182347, 6355119734595765, 11642138005593761, 28921424479420089, 55584853414001304, 183094184161456101
Offset: 1

Views

Author

James G. Merickel, Sep 19 2012

Keywords

Comments

See comments in A216840 and A216841. Smaller of pair of bases are 7, 9, 12, 15, 16, 19, 22, 23, 25, 28. The A216*** sequences in cross-reference are others in the collection including this one, while A171*** sequences are various record multi-base palindromes.

Examples

			The first term is 53363336335 in base 7 and is 13457775431 in base 8.

Crossrefs

Cf. A171701, A171702, A171703, A171704, A171705, A171706, A171740, A171741, A171742, A171775, A216840, A216841, A216843, A216899, A216900, A216901, A216902, A216903, A216904, A216906, A216907, A216908, A216909, A216910.

Extensions

a(11)-a(14) from Chai Wah Wu, Aug 19 2015

A216902 Smallest palindromic number of length 8 in two bases differing by n.

Original entry on oeis.org

2727014580, 526323690, 342324801, 1125252600, 2373480208, 128982894600, 72514770528, 47794089850, 91641847675, 300187699944, 292567324441, 2529713006600, 342239511285, 1287696341932
Offset: 2

Views

Author

James G. Merickel, Sep 19 2012

Keywords

Comments

See comments in A216840 and A216841. Smaller of pair of bases are 17, 14, 12, 14, 15, 27, 23, 24, 24, 32, 28, 36, 30, 36. The other sequences of this collection are in the A216*** cross-references, while the A171*** c-refs are various record multi-base palindromes.

Examples

			The first term is 6AGAAGA6 when written in base 17 and is 30I66I03 when written in base 19 (with notation analogous to that used in hexadecimal, A=10, B=11, et cetera).

Crossrefs

Cf. A171701, A171702, A171703, A171704, A171705, A171706, A171740, A171741, A171742, A171775, A216840, A216841, A216843, A216899, A216900, A216901, A216903, A216904, A216905, A216906, A216907, A216908, A216909, A216910.

A216903 Smallest palindromic number of 9 digits in two bases differing by n.

Original entry on oeis.org

27924649, 130535031, 952404259, 8755399858, 17207421669, 78641539396, 153703346251, 302665883230, 1141282588378, 1408219987879
Offset: 1

Views

Author

James G. Merickel, Sep 19 2012

Keywords

Comments

See comments to A216840 and A216841. Smaller bases of pair are 7, 8, 10, 13, 14, 17, 18, 19, 22, 23. Other sequences directly connected with this one are the A216*** cross-references, while the A171*** ones are various record multi-base palindromes.

Examples

			The first term is 456232654 in base 7 and is 152414251 in base 8.

Crossrefs

Cf. A171701, A171702, A171703, A171704, A171705, A171706, A171740, A171741, A171742, A171775, A216840, A216841, A216843, A216899, A216900, A216901, A216902, A216904, A216905, A216906, A216907, A216908, A216909, A216910.
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