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 1-4 of 4 results.

A116993 a(n) is the least number having exactly n representations as a product of two palindromes.

Original entry on oeis.org

13, 1, 4, 44, 66, 484, 4444, 7326, 6666, 48884, 73326, 493284, 888888, 666666, 5426124, 4888884, 6672666, 7333326, 44888844, 73399326, 246888642, 67333266, 4073662593, 4893772884, 4533773244, 6800659866, 2715775062, 1481331852, 493777284, 740665926, 8147325186, 5431550124, 74807258526
Offset: 0

Views

Author

Giovanni Resta, Apr 02 2006

Keywords

Comments

a(20) <= 733333326; a(34) <= 666666666666; a(39) <= 4888888888884 and a(44) <= 7333333333326. - Farideh Firoozbakht, Dec 10 2006

Examples

			a(0)=13 since 13 is the smallest number that cannot be represented as a product of two palindromes.
a(5)=484 since 484 = 1*484 = 2*242 = 4*121 = 22*22 = 11*44.

Links

Crossrefs

Cf. A002113, A125832, A125833, A125834, A140332.

Programs

  • Mathematica
    f[n_]:=f[n]=Length[Select[Divisors[n], #<=n^(1/2)&&FromDigits[ Reverse[IntegerDigits[ # ]]]==#&&FromDigits[Reverse[IntegerDigits [n/# ]]]==n/#&]]; a[n_]:=(For[m=1, f[m] != n, m++ ]; m); Do[Print[a[n]], {n, 0, 18}] (* Farideh Firoozbakht, Dec 10 2006 *)

Extensions

More terms from Farideh Firoozbakht, Dec 10 2006
a(19)-a(27) from Donovan Johnson, Aug 04 2009
More terms from David A. Corneth, Aug 10 2025

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

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 *)

A368955 Numbers that are the product of two repdigit numbers.

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, 110, 111, 121, 132, 154, 165, 176, 198, 220, 222, 231, 242, 264, 275, 297, 308, 330, 333
Offset: 1

Views

Author

Stefano Spezia, Jan 10 2024

Keywords

Comments

A010785 and A368944 are subsequences.
Numbers of the form i*j*(10^k - 1)*(10^m - 1)/81 where 0 <= i, j <= 9 and k, m >= 0.

Crossrefs

Cf. A002283, A010785, A140332, A368944.

Programs

  • Mathematica
    repQ[n_] := SameQ @@ IntegerDigits[n]; q[n_] := AnyTrue[Divisors[n], repQ[#] && repQ[n/#] &]; q[0] = True; Select[Range[0, 333], q] (* Amiram Eldar, Jan 12 2024 *)
  • Python
    from itertools import count, takewhile
def repdigits():
    yield 0
    yield from ((10**d-1)//9*i for d in count(1) for i in range(1, 10))
def aupto(LIMIT): # use LIMIT = 10**34 for 10K+-term b-file
    s, R = set(), list(takewhile(lambda x:x<=LIMIT, repdigits()))
    for i, r1 in enumerate(R):
        for r2 in R[i:]:
            p = r1*r2
            if p > LIMIT: break
            s.add(p)
    return sorted(s)
print(aupto(333)) # Michael S. Branicky, Jan 10 2024

Formula

a(n) = A140332(n) for n <= 46.
Showing 1-4 of 4 results.

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