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 1-5 of 5 results.

A328332 Expansion of (1 + 4*x - 5*x^2 + 10*x^3) / ((1 - x) * (1 - 10*x^2)).

Original entry on oeis.org

1, 5, 10, 60, 110, 610, 1110, 6110, 11110, 61110, 111110, 611110, 1111110, 6111110, 11111110, 61111110, 111111110, 611111110, 1111111110, 6111111110, 11111111110, 61111111110, 111111111110, 611111111110, 1111111111110, 6111111111110, 11111111111110, 61111111111110, 111111111111110
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 12 2019

Keywords

Comments

Number of odd palindromes <= 10^n.

Links

Crossrefs

Cf. A002113, A029950, A050250, A070199, A328333.

Programs

  • Mathematica
    nmax = 28; CoefficientList[Series[(1 + 4 x - 5 x^2 + 10 x^3) / ((1 - x) (1 - 10 x^2)), {x, 0, nmax}], x]
Join[{1}, LinearRecurrence[{1, 10, -10}, {5, 10, 60}, 28]]
  • PARI
    Vec((1 + 4*x - 5*x^2 + 10*x^3) / ((1 - x) * (1 - 10*x^2)) + O(x^30)) \\ Michel Marcus, Oct 13 2019

Formula

G.f.: (1 + 4*x - 5*x^2 + 10*x^3) / ((1 - x) * (1 - 10*x^2)).
a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3). - Wesley Ivan Hurt, Aug 25 2022

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.

A092361 Palindromic numbers containing one or more odd digits.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 33, 55, 77, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 212, 232, 252, 272, 292, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 414, 434, 454, 474, 494, 505, 515, 525, 535, 545, 555, 565, 575, 585, 595, 616, 636, 656, 676, 696
Offset: 1

Views

Author

Michael Joseph Halm, Mar 19 2004

Keywords

Comments

Begins to differ from the odd palindromic numbers, A029950, in the 21st term.

Examples

			a(21) = 212 because it is the 21st palindromic number with an odd digit, the first even one

Links

Crossrefs

Cf. A029950, A011539.

Programs

  • Mathematica
    Select[Range[1000],PalindromeQ[#]&&AnyTrue[IntegerDigits[#],OddQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 24 2021 *)

Extensions

Corrected and definition clarified by Harvey P. Dale, May 24 2021

A113578 a(1) = 1, then the rearrangement of odd palindromes such that every concatenation is a prime for n > 1.

Original entry on oeis.org

1, 3, 7, 11, 9, 111, 33, 99, 717, 151, 383, 969, 3003, 3663, 141, 121, 10101, 11711, 393, 11811, 363, 979, 77, 34443, 171, 14941, 989, 919, 707, 34243, 929, 7557, 18781, 18681, 131, 11511, 30303, 10701, 12421, 12321, 747, 7667, 1441, 14841, 13431, 797, 16861
Offset: 1

Views

Author

Amarnath Murthy, Nov 06 2005

Keywords

Comments

Since the first 5 terms of A083754 are odd palindromes (A029950), they are also the first 5 terms of this sequence. - Michel Marcus, Feb 06 2014

Examples

			13, 137, 13711, 137119, 137119111, 13711911133, ...,  are all prime.

Crossrefs

Cf. A113577, A113579.

Programs

  • PARI
    findnew(va, n, vp, ilast) = {s= ""; for (i=1, n-1, s = concat(s, Str(va[i]));); ok = 0; i = 2; while (!ok, if (vp[i] != 0, ns = concat(s, Str(vp[i])); if (isprime(eval(ns)), ok = 1);); if (!ok, i++); if (i > #vp, return (0));); i;}
lista(nn) = {vn = vector(nn, i, i); vp = select(n->is_A002113(n), vn); va = vector(nn); va[1] = 1; print1(va[1], ", "); vp[1] = 0; ilast = 1; for (n = 2, vecmax(vp), inew = findnew(va, n, vp, ilast); if (! inew, break); va[n] = vp[inew]; vp[inew] = 0; print1(va[n], ", "); ilast = inew;);} \\ Michel Marcus, Feb 06 2014

Extensions

Corrected and extended by Michel Marcus, Feb 06 2014

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 1-5 of 5 results.

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