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 16 results. Next

A140332 Products of two palindromes in base 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32, 33, 35, 36, 40, 42, 44, 45, 48, 49, 54, 55, 56, 63, 64, 66, 72, 77, 81, 88, 99, 101, 110, 111, 121, 131, 132, 141, 151, 154, 161, 165, 171, 176, 181, 191, 198
Offset: 1

Views

Author

Jonathan Vos Post, May 28 2008

Keywords

Comments

Geneviève Paquin, p. 5: "Lemma 3.7: a Christoffel word can always be written as the product of two palindromes."
Contains A115683 and A141322 as proper subsets.

Links

Crossrefs

Cf. A002113, A115683, A141322.

Programs

  • Maple
    digrev:= proc(n) local L,i; L:= convert(n,base,10); add(L[-i]*10^(i-1),i=1..nops(L)) end:
N:=3:
Res:= $0..9:
for d from 2 to N do
  if d::even then
    m:= d/2;
    Res:= Res, seq(n*10^m + digrev(n), n=10^(m-1)..10^m-1);
  else
    m:= (d-1)/2;
    Res:= Res, seq(seq(n*10^(m+1)+y*10^m+digrev(n), y=0..9), n=10^(m-1)..10^m-1);
  fi
od:
Palis:= [Res]:
Res:= 0:
for i from 2 to nops(Palis) while Palis[i]^2 <= 10^N do
  for j from i to nops(Palis) while Palis[i]*Palis[j] <= 10^N do
     Res:= Res, Palis[i]*Palis[j];
od od:sort(convert({Res},list)); # Robert Israel, Jan 06 2020
  • Mathematica
    pal = Select[ Range[0, 200], # == FromDigits@ Reverse@ IntegerDigits@ # &]; Select[ Union[ Times @@@ Tuples[pal, 2]], # <= 200 &] (* Giovanni Resta, Jun 20 2016 *)

Formula

A002113 UNION A115683.

Extensions

Data corrected by Giovanni Resta, Jun 20 2016

A115743 Squares that are the product of 2 palindromes greater than 1.

Original entry on oeis.org

4, 9, 16, 25, 36, 49, 64, 81, 121, 484, 1089, 1764, 1936, 2401, 2704, 3025, 4356, 5929, 6084, 7744, 9801, 10201, 12321, 14641, 17161, 19881, 22801, 25921, 27225, 29241, 32761, 36481, 40804, 44944, 49284, 53824, 58564, 63504, 68644, 69696
Offset: 1

Views

Author

Giovanni Resta, Jan 31 2006

Keywords

Comments

Are most terms of the form p^2 where p is a palindrome? - David A. Corneth, May 25 2021

Examples

			6084 = 78^2 and 6084 = 9*676.

Links

Crossrefs

Cf. A014186, A115683, A115744, A115745.

Programs

  • Mathematica
    With[{nn=50000},Select[Union[Select[Times@@@Tuples[Select[Range[2,nn],PalindromeQ],2],IntegerQ[ Sqrt[ #]]&]],#<=2 nn&]] (* Harvey P. Dale, Aug 21 2022 *)

A115744 Triangular numbers that are the product of 2 palindromes greater than 1.

Original entry on oeis.org

6, 10, 15, 21, 28, 36, 45, 55, 66, 231, 528, 666, 1128, 2016, 2211, 2628, 2775, 3003, 3570, 3916, 4095, 5995, 6105, 6216, 6903, 8646, 21736, 31878, 34980, 37950, 43956, 45753, 52003, 58653, 65703, 66066, 73153, 83028, 89676, 93528, 116886
Offset: 1

Views

Author

Giovanni Resta, Jan 31 2006

Keywords

Examples

			6903=T(117) and 6903=9*767.

Links

Crossrefs

Cf. A115683, A115743, A115745.

Programs

  • Mathematica
    Module[{upto=120000,pals},pals=Select[Range[2,upto/2],PalindromeQ];Select[ Times@@@Tuples[pals,2],OddQ[Sqrt[8#+1]]&&#<=upto&]]//Union (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 13 2018 *)

A141322 Nonpalindromes which are products of two palindromes in base 10.

Original entry on oeis.org

10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 45, 48, 49, 54, 56, 63, 64, 72, 81, 110, 132, 154, 165, 176, 198, 220, 231, 264, 275, 297, 302, 308, 322, 330, 342, 352, 362, 382, 385, 396, 423, 440, 453, 462, 483, 495, 504, 513, 524, 528
Offset: 1

Views

Author

Jonathan Vos Post, Aug 02 2008

Keywords

Examples

			726 is in this sequence because 22 * 33 = 726, 22 and 33 are palindromes base 10, but 726 is not a palindrome base 10.

Links

Crossrefs

Cf. A002113, A004086, A140332.

Programs

  • Maple
    digrev:= proc(n) local L,i; L:= convert(n,base,10); add(L[-i]*10^(i-1),i=1..nops(L)) end:
N:=3: # for terms of at most N digits
Res:= $0..9:
for d from 2 to N do
  if d::even then
    m:= d/2;
    Res:= Res, seq(n*10^m + digrev(n), n=10^(m-1)..10^m-1);
  else
    m:= (d-1)/2;
    Res:= Res, seq(seq(n*10^(m+1)+y*10^m+digrev(n), y=0..9), n=10^(m-1)..10^m-1);
  fi
od:
Palis:= [Res]:
Res:= NULL:
for i from 3 to nops(Palis) while Palis[i]^2 <= 10^N do
  for j from i to nops(Palis) while Palis[i]*Palis[j] <= 10^N do
     v:= Palis[i]*Palis[j];      if digrev(v) <> v then Res:= Res, v fi;
od od:sort(convert({Res},list)); # Robert Israel, Jan 06 2020

Formula

{A140332 INTERSECTION COMPLEMENT(A002113)} = {n in A115683 and n <> A004086(n)}.

Extensions

Extended beyond 330 by R. J. Mathar, Aug 09 2008

A115745 Pentagonal numbers (A000326) that are the product of 2 palindromes greater than 1.

Original entry on oeis.org

12, 22, 35, 176, 330, 2380, 2625, 2882, 3432, 3876, 4845, 8855, 9801, 12650, 14652, 15251, 21901, 26070, 52360, 61105, 66045, 84847, 111657, 137562, 156332, 197472, 230300, 254410, 260625, 270725, 287547, 295482, 351142, 359905
Offset: 1

Views

Author

Giovanni Resta, Jan 31 2006

Keywords

Examples

			66045 is the 210th pentagonal number and 66045=111*595.

Crossrefs

Cf. A000326, A115683, A115743, A115744.

A115697 Powerful(1) numbers (A001694) whose digit reversal is the product of 2 palindromes greater than 1.

Original entry on oeis.org

4, 8, 9, 27, 36, 72, 81, 121, 243, 324, 400, 484, 648, 800, 900, 1089, 1331, 1728, 1800, 2187, 2197, 2700, 3267, 3456, 3600, 4000, 4356, 4500, 5184, 5292, 5324, 5400, 5625, 6561, 7200, 7803, 8000, 8100, 8712, 9000, 9261, 9801, 10201, 11979
Offset: 1

Views

Author

Giovanni Resta, Jan 31 2006

Keywords

Examples

			3267=3^3*11^2 is powerful and 7623=77*99.

Links

Crossrefs

Cf. A001694, A115683.

A334140 Numbers that can be written as a product of distinct palindromes.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 27, 28, 30, 32, 33, 35, 36, 40, 42, 44, 45, 48, 54, 55, 56, 60, 63, 64, 66, 70, 72, 77, 80, 84, 88, 90, 96, 99, 101, 105, 108, 110, 111, 112, 120, 121, 126, 131, 132, 135, 140
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 15 2020

Keywords

Links

Crossrefs

Cf. A002113, A033620, A084034, A115683, A140332, A334131.

Programs

  • Mathematica
    ok[n_, w_: {}] := n <= 1 || AnyTrue[ Divisors@ n, ! MemberQ[w, #] && PalindromeQ[#] && ok[n/#, Append[w, #]] &]; Select[Range[0, 140], ok] (* Giovanni Resta, Apr 15 2020 *)

A115681 Brilliant numbers (A078972) whose digit reversal is the product of 2 palindromes greater than 1.

Original entry on oeis.org

4, 6, 9, 21, 121, 253, 407, 451, 559, 583, 667, 671, 803, 869, 2173, 2537, 5063, 5183, 5893, 10201, 13231, 15251, 16171, 18281, 19291, 22523, 22733, 24743, 25283, 26563, 27383, 28583, 28783, 31613, 35653, 37673, 38683, 40567, 45349, 46217
Offset: 1

Views

Author

Giovanni Resta, Jan 31 2006

Keywords

Examples

			22523=101*223 is brilliant and 32522=2*16261.

Crossrefs

Cf. A078972, A115683.

A115684 Both n and the reverse of n are the product of 2 palindromes greater than 1.

Original entry on oeis.org

4, 6, 8, 9, 12, 18, 21, 22, 24, 27, 33, 36, 40, 42, 44, 45, 54, 55, 63, 66, 72, 77, 81, 88, 99, 121, 132, 198, 202, 220, 222, 231, 242, 262, 264, 282, 297, 303, 330, 333, 363, 393, 396, 404, 424, 440, 444, 462, 464, 484, 495, 505, 555, 594, 606, 616, 626, 636
Offset: 1

Views

Author

Giovanni Resta, Jan 31 2006

Keywords

Examples

			264=6*11 and 462=7*66

Crossrefs

Cf. A002113, A115683.

A115698 Semiprimes (A001358) whose digit reversal is the product of 2 palindromes greater than 1.

Original entry on oeis.org

4, 6, 9, 21, 22, 33, 46, 51, 55, 65, 77, 82, 94, 121, 202, 203, 253, 262, 303, 393, 407, 427, 445, 446, 451, 485, 505, 559, 583, 626, 667, 669, 671, 687, 707, 789, 803, 849, 869, 889, 939, 1042, 1111, 1441, 1622, 1661, 1991, 2031, 2123, 2157, 2173, 2321
Offset: 1

Views

Author

Giovanni Resta, Jan 31 2006

Keywords

Examples

			669=3*223 is semiprime and 966=6*161.

Crossrefs

Cf. A001358, A115683.
Showing 1-10 of 16 results. Next

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