A171705 -id:A171705 - cp's OEIS frontend

A171702 The first number that is (at least) n-fold intrinsically 3-palindromic (represented in base ten).

5, 26, 121, 300, 1009, 1477, 4591, 15049, 35101, 35101, 83161, 176401, 291721, 291721, 720721, 1081081, 1496881, 2106721, 3341521, 3341521, 5654881, 7207201, 8648641, 10810801, 13693681, 15135121, 21621601, 28274401, 28274401, 41081041, 41081041, 41081041

Offset: 1

James G. Merickel, Dec 15 2009

			a(1)=5 is 101 in base 2.
a(2)=26 is 222 in base 3 and 101 in base 5.
a(3)=121 is 232 in base 7 and 171 in base 8.

Giovanni Resta, Table of n, a(n) for n = 1..100 [terms a(1)-a(38) computed by Hiroaki Yamanouchi; a(39)-a(100) by _Giovanni Resta_, Apr 22 2016]

Cf. A171701, A171703, A171704, A171705, A171706, A171740, A171741, A171742.

a(21)-a(32) from Hiroaki Yamanouchi, Sep 20 2014

A171704 The first n-fold intrinsically 5-palindrome (represented in base ten).

Original entry on oeis.org

17, 2293, 267140, 4838419019

Offset: 1

James G. Merickel, Dec 17 2009

The fourth term, which is first 5-palindromic in base 91, is almost a concatenation of base-ten 5-palindromes.
A217146 gives the sequence of values 5-palindromic in at least 4 bases.  The search there indicates a fifth term here has first base of 5-palindromicity above 200. - James G. Merickel Oct 19 2012
a(5) > 10^14. - Hiroaki Yamanouchi, Aug 22 2015

			a(2)=2293 is 33133 in base 5 and 14341 in base 6;
a(3)=267140 is 6D4D6, 54245 and 29E92 in bases 14, 15 and 18, respectively.

Cf. A171701, A171702, A171703, A171705, A171706, A171740, A171741, A171742 A217146.

A171706 The first n-fold intrinsically 7-palindromic number (represented in base 10).

Original entry on oeis.org

65, 186621, 3360633

Offset: 1

James G. Merickel, Dec 16 2009

Though this sequence is apt to never have many known members, its third is quite a nice coincidence for consideration.
A029965 runs just short of space to show it, and A053780 is a totally unrelated collection of palindromes that also includes 33633 in addition to 3360633.
a(4) > 10^17. - Hiroaki Yamanouchi, Aug 22 2015

			a(2)=186621 is 3555553 in base 6 and 1405041 in base 7.
a(3)=3360633 is 6281826 in base 9 and 1995991 in base 11.

Cf. A171701, A171702, A171703, A171704, A171705, A171741, A029966, A029965, A053780.

Typo in third term corrected by James G. Merickel, Dec 18 2009

A171701 The first n-fold (at least) intrinsically 2-palindromic number (in base ten).

Original entry on oeis.org

3, 8, 18, 24, 48, 60, 120, 120, 180, 336, 360, 360, 720, 720, 720, 840, 1260, 1440, 1680, 1680, 2520, 2520, 2520, 2520, 5040, 5040, 5040, 5040, 5040, 5040, 7560, 7560, 10080, 10080, 10080, 10080, 15120, 15120, 15120, 15120, 30160, 20160, 25200, 25200

Offset: 1

James G. Merickel, Dec 16 2009

Cf. A171702, A171703, A171704, A171705, A171706, A171740, A171741, A171742.

If d'(k) is the number of divisors of k minus 1 for square k and minus 2 for numbers of the form m^2+m, then a(n) is the first k for which d'(k)/2>n-1.

A171703 The first n-fold intrinsically 4-palindromic number (represented in base ten).

Original entry on oeis.org

9, 624, 19040, 30222192, 592704000, 12040481088, 128024064000, 1024192512000, 3456649728000

Offset: 1

James G. Merickel, Dec 16 2009

The fifth and sixth terms are both cubes, exploiting the simple binomial expansion as (m)(m)^3=(m^3)(3m^3)(3m^3)(m^3) for much of their status. - James G. Merickel, Dec 19 2009
The first base in which the fifth term is 4-palindromic is 209. - James G. Merickel, Dec 18 2009
a(10) <= 27653197824000, a(11) <= 27653197824000, a(12) <= 575100098496000, a(13) <= 1733189185728000, a(14) <= 1733189185728000, a(15) <= 8400090327552000. - Hiroaki Yamanouchi, Sep 24 2014

			a(2)=624 is 4444 in base 5 and 1551 in base 7.

Cf. A171701, A171702, A171704, A171705, A171706, A171740, A171741, A171742.

a(5)-a(6) from James G. Merickel, Dec 19 2009

a(7)-a(9) from Hiroaki Yamanouchi, Sep 24 2014

A171741 The first 3-fold intrinsically n-palindromic number (given in base ten).

Original entry on oeis.org

18, 154, 19040, 267140, 2853326516320, 3360633

Offset: 2

James G. Merickel, Dec 17 2009

The large value for the 6-palindrome case, a(6)=2853326516320, has 139 as the smallest of the three bases.
Coincidentally, the following much shorter term, a(7)= 3360633, involves base 9 through 11, the conjunction of base-9 and base-10 palindromes (A029965) comes up a term short (assuming space limitations remain as they are at the beginning of 2011), and A053740 is a totally unrelated collection of palindromes that also contains 33633.
a(8) > 10^18. - Hiroaki Yamanouchi, Aug 22 2015

			a(5)=267140 is 6D4D6 in base 14, 54245 in base 15, and 29E92 in base 18.
a(7)=3360633 is 6281826 in base 9 and 1995991 in base 11.

Cf. A171701, A171702, A171703, A171704, A171705, A171706, A171740, A171742, A053780, A029966, A029965.

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

James G. Merickel, Dec 17 2009

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

			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.

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

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

A171742 First 4-fold intrinsically n-palindrome (given in base ten).

Original entry on oeis.org

24, 300, 30222192, 4838419019

Offset: 2

James G. Merickel, Dec 17 2009

Given the size of the first 3-fold 6-palindrome (13 base-ten digits; see A171705 or A171741), a fifth term is going to be difficult.

			a(2)=24 is 11 in base 23, 22 in base 11, 33 in base 7 and 44 in base 5.
a(3)=300 is 1A1 in base 13, 363 in base 9, 454 in base 8 and 606 in base 7.

Cf. A171701, A171702, A171703, A171704, A171705, A171706, A171740, A171741.

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

James G. Merickel, Sep 19 2012

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.

			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

Chai Wah Wu, Table of n, a(n) for n = 1..10000

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.

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

James G. Merickel, Sep 19 2012

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.

			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

Chai Wah Wu, Table of n, a(n) for n = 2..10000

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.

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

cp's OEIS Frontend

A171702 The first number that is (at least) n-fold intrinsically 3-palindromic (represented in base ten).

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A171704 The first n-fold intrinsically 5-palindrome (represented in base ten).

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A171706 The first n-fold intrinsically 7-palindromic number (represented in base 10).

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A171701 The first n-fold (at least) intrinsically 2-palindromic number (in base ten).

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A171703 The first n-fold intrinsically 4-palindromic number (represented in base ten).

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A171741 The first 3-fold intrinsically n-palindromic number (given in base ten).

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

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A171742 First 4-fold intrinsically n-palindrome (given in base ten).

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

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

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