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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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

A171775 a(n) = smallest number M such that there exist bases b_2, b_3, ..., b_n with the property that M written in base b_k is a k-digit palindrome for all k=2..n.

Original entry on oeis.org

1, 3, 5, 52, 130, 1885, 1073741824, 4398046511104
Offset: 1

Views

Author

James G. Merickel, Dec 18 2009

Keywords

Comments

a(n) is no more than 2^[(n-1)*(n-2)] for n > 6 (and equals it for n = 7 and 8 at least). The reason for this bound is that for this number for each length from n down to 3 there is at least one power of 2, 2^k, such that in base b = 2^k-1 the binomial expansion of (b+1)^floor([(n-1)*(n-2)]/k) multiplied by the remaining small power of 2 gives a palindromic expression not requiring carries in base b. James G. Merickel, Aug 05 2015

Examples

			a(6)=1885: the bases are 1884 (1885 is 11 in base 1884), 14 (1885 is 989 in base 14), 12 (it is 1111 in base 12), 6 (it is 12421 in base 6), and 4 (it is 131131 in base 4).

Crossrefs

Cf. A007632, A007633, A029961, A029962, A029963, A029964, A029804, A029965, A029966, A029967, A029968, A029969, A029970, A029731, A097855, A099165, A087155, A171701, A171702, A171703, A171704, A171705, A171706, A171740, A171741, A171742, A253294.

Extensions

a(7) and a(8) added by James G. Merickel, Feb 04 2010
Offset changed to 1, with corresponding addition of a(1) by James G. Merickel, Jul 24 2015
Comment corrected and explained.James G. Merickel, Aug 05 2015
Definition and example rewritten by N. J. A. Sloane, Aug 05 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.

A216904 Smallest palindromic number of length 10 in two bases differing by n.

Original entry on oeis.org

260311602096, 1161970522902, 3240922809120, 16446205929150, 5681121663231, 86430674537922, 36434748011040, 2119208930874272, 1198501397192925, 850466252340408, 567228097076960, 80847888446187460, 5733768413240616, 12410554422030840, 3028257303808673, 32977836735154716, 27000016202473335, 239720296892530428, 57796746638482600
Offset: 2

Views

Author

James G. Merickel, Sep 19 2012

Keywords

Comments

See comments in A216840 and A216841. Smaller of pair of bases are 15, 17, 19, 24, 20, 25, 23, 36, 34. The A216*** cross-references are other sequences in the collection that includes this one, while A171*** sequences are various record multi-base palindromes.

Examples

			The first term is 6B882288B6 when written in base 15 and is 2356886532 when written in base 17.

Crossrefs

Cf. A171701, A171702, A171703, A171704, A171705, A171706, A171740, A171741, A171742, A171775, A216840, A216841, A216843, A216899, A216900, A216901, A216902, A216903, A216905, A216906, A216907, A216908, A216909, A216910.

Extensions

a(11)-a(20) from Chai Wah Wu, Aug 19 2015

A216906 Smallest palindromic number of length 12 in two bases differing by n.

Original entry on oeis.org

248876637484140, 1013960892216384, 1093158208940760, 101975569769446364, 16699985993414520, 1310809863419914830, 331113952227684640
Offset: 2

Views

Author

James G. Merickel, Sep 19 2012

Keywords

Comments

See comments in A216840, A216841. Smaller of pair of bases are 17, 20, 19, 28, 23. A216*** cross-references are other sequences in a collection that includes this one, while A171*** sequences are various record multi-base palindromes.

Examples

			The first term is 747B6336B747 in base 17 and 22B4I88I4B22 in base 19.

Crossrefs

Cf. A171701, A171702, A171703, A171704, A171705, A171706, A171740, A171741, A171742, A171775, A216840, A216841, A216843, A216899, A216900, A216901, A216902, A216903, A216904, A216905, A216907, A216908, A216909, A216910.

Extensions

a(7)-a(8) from Chai Wah Wu, Aug 19 2015

A216907 Smallest palindromic number of length 13 in two bases differing by n.

Original entry on oeis.org

318713056300, 101252880671531, 778061547945037, 8047782883062742, 37866346844745025, 247644110790746909, 828823170233185174, 10556938000726441969, 10175119610486126649, 34575397981277771773, 91506808775956056797
Offset: 1

Views

Author

James G. Merickel, Sep 19 2012

Keywords

Comments

See comments sections of A216840 and A216841. Smallest of pair of bases are 8, 12, 14, 17, 19, 22, 24, 30, 32. For the remarkable case a(10), not only does the number have four 7s in a row in base 10, but it also has prime reversal there and in its 2nd base of 13-palindromicity, 42, no digit larger than 20 appears. A216*** cross-references are others in a collection of sequences that includes this one, while A171*** ones are various record multi-base palindromes.

Examples

			The first term is 4506461646054 in base 8 and 1113578753111 in base 9.

Crossrefs

Cf. A171701, A171702, A171703, A171704, A171705, A171706, A171740, A171741, A171742, A171775, A216840, A216841, A216843, A216899, A216900, A216901, A216902, A216903, A216904, A216905, A216906, A216908, A216909, A216910.

Extensions

a(10) added by James G. Merickel, Oct 17 2012
a(11) from Chai Wah Wu, Aug 20 2015
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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