A052063 -id:A052063 - cp's OEIS frontend

A052061 Numbers k such that decimal expansion of k^2 contains no palindromic substring except single digits.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 16, 17, 18, 19, 23, 24, 25, 27, 28, 29, 31, 32, 33, 36, 37, 39, 41, 42, 43, 44, 48, 49, 51, 52, 53, 54, 55, 57, 59, 61, 64, 66, 68, 69, 71, 72, 73, 74, 75, 78, 79, 82, 84, 86, 87, 89, 93, 95, 96, 97, 98, 99, 104, 113, 116, 117, 118, 124

Offset: 1

Patrick De Geest, Jan 15 2000

Leading zeros in the substrings are allowed so 103^2 = 10609 is rejected because 1{060}9 contains a palindromic substring.

Probabilistic analysis strongly suggests that this sequence is not finite. - Franklin T. Adams-Watters, Nov 15 2006

			118^2 = 13924 -> substrings 13, 39, 92, 24, 139, 392, 924, 1392, 3924 and 13924 are all non-palindromic.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10001

Cf. A052062, A052063, A052064, A050741.

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=0,200,if(noPalSub(n^2),print1(n", ")))

Program and b-file from Charles R Greathouse IV, Sep 09 2009

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

Patrick De Geest, Jan 15 2000

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.

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

Charles R Greathouse IV, Table of n, a(n) for n = 1..10001

Cf. A052061, A052063, A052064, A050749.

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", ")))

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

Program and b-file from Charles R Greathouse IV, Sep 09 2009

Offset changed by Andrew Howroyd, Aug 11 2024

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

Patrick De Geest, Jan 15 2000

Sequence is probably finite.

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

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

Harvey P. Dale, Table of n, a(n) for n = 1..1001

Cf. A052063, A052061, A052062, A050750.

npsQ[n_]:=Count[Flatten[Table[Partition[IntegerDigits[n],i,1],{i,2, IntegerLength[ n]}],1],?(#==Reverse[#]&)]==0; Select[Range[ 0,100]^3, npsQ] (* _Harvey P. Dale, Dec 10 2016 *)

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

A052067 a(n) is the first cube root greater than 10^n such that a(n)^3 is a palfree cube (palfree = contains no palindromic substring except single digits).

Original entry on oeis.org

12, 102, 1008, 10091, 100698, 1007059, 10069605, 100695969, 1006958659, 10069585741, 100695847434, 1006958474563, 10069584743393, 100695847434688, 1006958474332019, 10069584743315203

Offset: 1

Patrick De Geest, Jan 15 2000

Probably finite.

Keith Schneider, Table of n, a(n) for n = 1..25
Keith Schneider, Mathematica code for A052025-A052028

Cf. A052068, A052063, A052064.

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

A052068 Palfree cubes whose root is the smallest possible greater than 10^n (palfree = contains no palindromic substring except single digits).

Original entry on oeis.org

1728, 1061208, 1024192512, 1027549183571, 1021086501268392, 1021326840189306379, 1021027182906953620125, 1021024718963175618538209, 1021021582763927831895785179

Offset: 1

Patrick De Geest, Jan 15 2000

Probably finite.

Keith Schneider, Table of n, a(n) for n = 1..25
Keith Schneider, Mathematica code for A052025-A052028

Cf. A052067, A052063, A052064.

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

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A052061 Numbers k such that decimal expansion of k^2 contains no palindromic substring except single digits.

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A052062 Squares containing no palindromic substring except single digits.

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A052064 Cubes containing no palindromic substring except single digits.

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A052067 a(n) is the first cube root greater than 10^n such that a(n)^3 is a palfree cube (palfree = contains no palindromic substring except single digits).

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A052068 Palfree cubes whose root is the smallest possible greater than 10^n (palfree = contains no palindromic substring except single digits).

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