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 1-7 of 7 results.

A344422 Palindromes having more divisors than all smaller palindromes.

Original entry on oeis.org

1, 2, 4, 6, 44, 66, 252, 2112, 2772, 6336, 27972, 48384, 219912, 252252, 696696, 828828, 2114112, 4228224, 21333312, 42666624, 63999936, 234666432, 2154664512, 2329559232, 4815995184, 8402442048, 21354645312, 40362626304, 63708380736, 211887788112
Offset: 1

Views

Author

Bhupendra Kumar Singh, May 17 2021

Keywords

Comments

A000005(a(n)) = 1, 2, 3, 4, 6, 8, 18, 28, 36, 42, 48, 72, 96, 108, 128, 144, 168, 192, 336, 384, .... - Felix Fröhlich, May 19 2021
From Jon E. Schoenfield, Jun 22 2021: (Start)
There exists at least one m-digit term for every m in 1..22 except 21 (see the b-file).
Conjecture: all terms after a(1)=1 are even. (End)

Examples

			Terms include: 4 (3 divisors); 6 (4 divisors); 44 (6 divisors); 66 (8 divisors); 252 (18 divisors).

Links

Crossrefs

Cf. A000005, A002113 (palindromes), A076888 (their number of divisors), A002182, A084324, A093036, A345250.

Programs

  • Mathematica
    pal=Union@Flatten[Table[r=IntegerDigits@n;FromDigits/@(Join[r,#]&/@{Reverse@r,Rest@Reverse@r}),{n,10^5}]];m=0;lst={};Do[s=DivisorSigma[0,k];If[s>m,AppendTo[lst,k];m=s],{k,pal}];lst (* Giorgos Kalogeropoulos, Dec 08 2021 *)
  • PARI
    \\ See PARI link. David A. Corneth, May 18 2021

Formula

A000005(a(n)) = A345250(n).

Extensions

Data corrected and extended by David A. Corneth, May 18 2021

A345250 a(n) is the number of divisors of A344422(n).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 18, 28, 36, 42, 48, 72, 96, 108, 128, 144, 168, 192, 336, 384, 448, 504, 672, 896, 960, 1008, 1134, 1296, 1344, 2560, 2592, 3456, 3584, 4320, 4608, 5376, 6144, 6336, 6912, 9216, 10240, 12288, 13824, 15360, 16128, 16384, 20736, 23328, 24576
Offset: 1

Views

Author

Bhupendra Kumar Singh, Jun 12 2021

Keywords

Comments

Sequence proposed by Felix Fröhlich in the first comment of A344422.

Links

Crossrefs

Records in A076888.
Cf. A000005, A002113, A002182, A344422.

Formula

a(n) = A000005(A344422(n)).

A083867 a(n) is the number of divisors of the n-th decimal palindrome that are palindromes.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 2, 4, 4, 6, 4, 8, 4, 8, 6, 2, 3, 3, 2, 3, 2, 3, 4, 2, 2, 4, 4, 6, 5, 6, 8, 4, 5, 6, 4, 4, 2, 2, 5, 3, 2, 6, 2, 2, 4, 6, 6, 6, 4, 8, 3, 6, 5, 9, 3, 4, 3, 5, 3, 3, 5, 3, 3, 5, 4, 8, 11, 4, 7, 4, 5, 9, 4, 5, 8, 4, 3, 2, 3, 4, 2, 2, 5, 2, 2, 8, 3, 8, 3, 7, 9, 6, 3, 10, 3, 6, 2, 2, 4, 2, 3, 4
Offset: 1

Views

Author

Reinhard Zumkeller, May 07 2003

Keywords

Examples

			n=45, divisors of A002113(45)=363 are {1,3,11,33,121,363}, all are palindromes, therefore a(45)=A076888(45)=6.
n=72, divisors of A002113(72)=636 are {1,2,3,4,6,12,53,106,159,212,318,636}, 7 of them are palindromes {1,2,3,4,6,212,636}, therefore a(72)=7 < A076888(72)=12.

Crossrefs

Cf. A083868, A083870, A076888, A000005, A002113.

Programs

A343735 Odd palindromes having more divisors than all smaller odd palindromes.

Original entry on oeis.org

1, 3, 9, 33, 99, 525, 3003, 5445, 5775, 50505, 53235, 171171, 525525, 5073705, 18999981, 50555505, 51666615, 512272215, 513828315, 5026226205, 5053553505, 5184994815, 5708778075, 52252425225, 502299992205, 502875578205, 524241142425, 579024420975
Offset: 1

Views

Author

Jon E. Schoenfield, Jun 22 2021

Keywords

Comments

A000005(a(n)) = A343736(n).
Conjectures:
(1) All terms after a(1)=1 are multiples of 3.
(2) The number of terms after a(30)=34418522581443 that are not multiples of 5 is finite but not zero.

Examples

			                                                      no. of
   n        a(n)  prime factorization                divisors
  --  ----------  ---------------------------------  --------
   1           1  -                                         1
   2           3  3                                         2
   3           9  3^2                                       3
   4          33  3 * 11                                    4
   5          99  3^2 * 11                                  6
   6         525  3 * 5^2 * 7                              12
   7        3003  3 * 7 * 11 * 13                          16
   8        5445  3^2 * 5 * 11^2                           18
   9        5775  3 * 5^2 * 7 * 11                         24
  10       50505  3 * 5 * 7 * 13 * 37                      32
  11       53235  3^2 * 5 * 7 * 13^2                       36
  12      171171  3^2 * 7 * 11 * 13 * 19                   48
  13      525525  3 * 5^2 * 7^2 * 11 * 13                  72
  14     5073705  3^3 * 5 * 7^2 * 13 * 59                  96
  15    18999981  3^3 * 7 * 11 * 13 * 19 * 37             128
  16    50555505  3 * 5 * 7^2 * 11 * 13^2 * 37            144
  17    51666615  3^2 * 5 * 7 * 11 * 13 * 31 * 37         192
  18   512272215  3^3 * 5 * 7^3 * 13 * 23 * 37            256
  19   513828315  3^2 * 5 * 7 * 11^2 * 13 * 17 * 61       288
  20  5026226205  3 * 5 * 7^2 * 11 * 13 * 17 * 29 * 97    384

Links

Crossrefs

Cf. A000005, A002113 (palindromes), A076888 (their number of divisors), A029950 (odd palindromes), A344422, A345250, A343736.

A083753 Smallest palindromic number with exactly n divisors, or 0 if no such number exists.

Original entry on oeis.org

1, 2, 4, 6, 14641, 44, 0, 66, 484, 272, 0, 414, 0, 2912192, 44944, 616, 0, 252, 0, 2992, 0, 2532352, 0, 4004, 10004000600040001, 2977792, 1002001, 2112, 0, 63536, 0, 4224, 0, 44356665344, 0, 2772, 0, 6564989894656, 0, 42224, 0, 6336, 0, 4015104, 698896
Offset: 1

Views

Author

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 06 2003

Keywords

Comments

a(7)=a(11)=a(13)=a(17)=a(19)=a(23)=a(29)=a(31)=a(37)=a(41)=0 under the plausible conjecture that there are no palindromes > 1 which are fifth or higher powers. David Wasserman in A090315 reports that he has checked this (or rather the part needed for this sequence) up to 10^48. - David Consiglio, Jr. and Charles R Greathouse IV, Mar 27 2012
a(21), a(33), a(35), and a(39) have also not been proved to be zero, but if positive they must be at least 10^31. - Charles R Greathouse IV, Mar 27 2012

Crossrefs

Cf. A002113, A076888.

Extensions

a(11)-a(42) from David Consiglio, Jr. and Charles R Greathouse IV, Mar 27 2012
a(43)-a(45) added (with a(43)=0 under the same conjecture as for a(7)=a(11)=...=a(41)=0) by Jon E. Schoenfield, Oct 17 2014

A083868 Number of palindromic divisors d of n-th decimal palindrome m, such that 9

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 1, 3, 1, 3, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 1, 2, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 5, 1, 1, 1, 0, 3, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 3, 0, 1, 0, 2, 4, 1, 0, 3, 0, 2, 0, 0, 1, 0, 0, 1
Offset: 1

Views

Author

Reinhard Zumkeller, May 07 2003

Keywords

Examples

			n=116, divisors of A002113(116) = 1771 are
{1,7,11,23,77,161,253,1771}, 6 of them are palindromes {1,7,11,77,161,1771}
and three are >9 and <1771: {11,77,161}, therefore
a(116) = 3 < A083867(116) = 6 < A076888(116) = 8.

Crossrefs

Cf. A078337, A083867, A076888, A000005, A002113.

A343736 a(n) is the number of divisors of A343735(n).

Original entry on oeis.org

1, 2, 3, 4, 6, 12, 16, 18, 24, 32, 36, 48, 72, 96, 128, 144, 192, 256, 288, 384, 432, 512, 576, 648, 768, 1024, 1296, 1536, 1728, 2048, 2304, 2592, 3456, 3888, 4608, 5760, 6912, 7680, 8640, 9216, 12288, 13824, 15360, 17280, 18432, 20480, 23040, 30720, 34560
Offset: 1

Views

Author

Jon E. Schoenfield, Jun 22 2021

Keywords

Crossrefs

Cf. A000005, A002113 (palindromes), A076888 (their number of divisors), A029950 (odd palindromes), A343735, A344422, A345250.

Formula

a(n) = A000005(A343735(n)).
Showing 1-7 of 7 results.

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