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 1-6 of 6 results.

A052062 Squares containing no palindromic substring except single digits.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 169, 196, 256, 289, 324, 361, 529, 576, 625, 729, 784, 841, 961, 1024, 1089, 1296, 1369, 1521, 1681, 1764, 1849, 1936, 2304, 2401, 2601, 2704, 2809, 2916, 3025, 3249, 3481, 3721, 4096, 4356, 4624, 4761, 5041, 5184, 5329
Offset: 1

Views

Author

Patrick De Geest, Jan 15 2000

Keywords

Comments

Leading zeros in substring allowed so 103^2 = 10609 is rejected because 1{060}9 contains a palindromic substring.
A comment in A052061 suggests that this sequence is infinite.

Examples

			2304 (= 48^2) -> substrings 23, 30, 04, 230, 304 and 2304 are all non-palindromic.

Links

Crossrefs

Cf. A052061, A052063, A052064, A050749.

Programs

  • PARI
    noPalSub(n)={my(d);local(digit);digit=eval(Vec(Str(n)));d = #digit;for(len=2,d,for(i=1,d-len+1,if(isPalSub(i,len), return(0))));1};isPalSub(start,len)={my(b=start-1,e=start+len);for(j=1,len>>1,if(digit[b+j] != digit[e-j], return(0)));1}
for(n=0,100,if(noPalSub(n^2),print1(n^2", ")))

Formula

a(n) = A052061(n)^2. - Andrew Howroyd, Aug 11 2024

Extensions

Program and b-file from Charles R Greathouse IV, Sep 09 2009
Offset changed by Andrew Howroyd, Aug 11 2024

A052063 Numbers k such that the decimal expansion of k^3 contains no palindromic substring except single digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 12, 13, 16, 17, 18, 19, 21, 22, 24, 25, 27, 28, 29, 32, 33, 35, 37, 38, 39, 41, 43, 44, 47, 51, 57, 59, 65, 66, 69, 73, 75, 76, 84, 88, 93, 94, 97, 102, 108, 109, 115, 116, 123, 125, 128, 133, 134, 135, 139, 144, 145, 147, 148, 155, 156, 159
Offset: 1

Views

Author

Patrick De Geest, Jan 15 2000

Keywords

Comments

Leading zeros in substring are allowed so 52^3 = 140608 is rejected because 14{060}8 contains a palindromic substring.
Probabilistic analysis strongly suggests that this sequence is not finite. - Franklin T. Adams-Watters, Nov 15 2006

Examples

			19^3 = 6859 -> substrings 68, 85, 59, 685, 859 and 6859 are all non-palindromic.

Links

Crossrefs

Cf. A052064, A052061, A052062, A050742.

Programs

  • Mathematica
    testQ@l_ :=
NoneTrue[Flatten[Table[Partition[l, n, 1], {n, 2, Length@l}], 1],
  PalindromeQ];
f@nn_ := Select[Range@nn, testQ@IntegerDigits@(#^3) &]; f[300]
(* Hans Rudolf Widmer, May 13 2022 *)
  • Python
    def nopal(s): return all(ss != ss[::-1] for ss in (s[i:j] for i in range(len(s)-1) for j in range(i+2, len(s)+1)))
def ok(n): return nopal(str(n**3))
print([k for k in range(160) if ok(k)]) # Michael S. Branicky, May 13 2022

A052064 Cubes containing no palindromic substring except single digits.

Original entry on oeis.org

0, 1, 8, 27, 64, 125, 216, 512, 729, 1728, 2197, 4096, 4913, 5832, 6859, 9261, 10648, 13824, 15625, 19683, 21952, 24389, 32768, 35937, 42875, 50653, 54872, 59319, 68921, 79507, 85184, 103823, 132651, 185193, 205379, 274625, 287496, 328509
Offset: 1

Views

Author

Patrick De Geest, Jan 15 2000

Keywords

Comments

Sequence is probably finite.
Leading zeros in substring allowed so 52^2 = 140608 is rejected because 14{060}8 contains a palindromic substring.

Examples

			132651 (= 51^3) -> substrings 13, 32, 26, 65, 51, 132, 326, 265, 651, 1326, 3265, 2651, 13265, 32651 and 132651 are all non-palindromic.

Links

Crossrefs

Cf. A052063, A052061, A052062, A050750.

Programs

Extensions

Offset changed to 1 by Sean A. Irvine, Oct 17 2019

A052065 a(n) is the first square root greater than 10^n such that a(n)^2 is a palfree square (palfree = contains no palindromic substring except single digits).

Original entry on oeis.org

13, 104, 1014, 10123, 101047, 1010456, 10104574, 101045587, 1010455851, 10104558492, 101045584913, 1010455848322, 10104558481373, 101045584813152, 1010455848130452, 10104558481304484
Offset: 1

Views

Author

Patrick De Geest, Jan 15 2000

Keywords

Comments

Probably finite.

Links

Crossrefs

Cf. A052066, A052061, A052062.

Extensions

More terms from Keith Schneider (schneidk(AT)email.unc.edu), May 23 2007

A052066 Palfree squares whose root is the smallest possible greater than 10^n (palfree = contains no palindromic substring except single digits).

Original entry on oeis.org

169, 10816, 1028196, 102475129, 10210496209, 1021021327936, 102102415721476, 10210210652174569, 1021021026820134201, 102102102318249314064, 10210210230410293217569
Offset: 1

Views

Author

Patrick De Geest, Jan 15 2000

Keywords

Comments

Probably finite.

Links

Crossrefs

Cf. A052065, A052061, A052062.

Extensions

More terms from Keith Schneider (schneidk(AT)email.unc.edu), May 23 2007

A151997 a(n) = (n 1's followed by a 3)^2.

Original entry on oeis.org

9, 169, 12769, 1238769, 123498769, 12346098769, 1234572098769, 123456832098769, 12345679432098769, 1234567905432098769, 123456790165432098769
Offset: 0

Views

Author

Zak Seidov, Sep 09 2009

Keywords

Comments

Is it true that the decimal expansion of a(n) contains no palindromic substrings of length greater than one?

Examples

			{3.9}
{13,169}
{113,12769}
{1113,1238769}
{11113,123498769}
{111113,12346098769}
{1111113,1234572098769}
{11111113,123456832098769}
{111111113,12345679432098769}
{1111111113,1234567905432098769}
{11111111113,123456790165432098769}

Crossrefs

Cf. A052061, A052062.

Programs

  • Mathematica
    Table[FromDigits[PadLeft[{3},n,1]]^2,{n,20}] (* Harvey P. Dale, Sep 04 2022 *)

Formula

a(n) = (100^(n+1)+340*10^n+289)/81. a(n)= 111*a(n-1) -1110*a(n-2) +1000*a(n-3). G.f.: (9-830*x+4000*x^2)/((1-x) * (100*x-1) * (10*x-1)). [From R. J. Mathar, Sep 15 2009]
Showing 1-6 of 6 results.

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