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.

A035090 Non-palindromic squares which when written backwards remain square (and still have the same number of digits).

Original entry on oeis.org

144, 169, 441, 961, 1089, 9801, 10404, 10609, 12544, 12769, 14884, 40401, 44521, 48841, 90601, 96721, 1004004, 1006009, 1022121, 1024144, 1026169, 1042441, 1044484, 1062961, 1212201, 1214404, 1216609, 1236544, 1238769, 1256641
Offset: 1

Views

Author

Patrick De Geest, Nov 15 1998

Keywords

Comments

Squares with trailing zeros not included.
Sequence is infinite, since it includes, e.g., 10^(2k) + 4*10^k + 4 for all k. - Robert Israel, Sep 20 2015

Links

Crossrefs

Reversing a polytopal number gives a polytopal number:
cube to cube: A035123, A035124, A035125, A002781;
square to square: A161902, A035090, A033294, A106323, A106324, A002779;
square to triangular: A181412, A066702;
tetrahedral to tetrahedral: A006030;
triangular to square: A066703, A179889;
triangular to triangular: A066528, A069673, A003098, A066569.
Cf. A319388.

Programs

  • Maple
    rev:= proc(n) local L,i;
L:= convert(n,base,10);
add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
filter:= proc(n) local t;
  if n mod 10 = 0 then return false fi;
  t:= rev(n);
t <> n and issqr(t)
end proc:
select(filter, [seq(n^2, n=1..10^5)]); # Robert Israel, Sep 20 2015
  • Mathematica
    Select[Range[1200]^2,!PalindromeQ[#]&&IntegerLength[#]==IntegerLength[ IntegerReverse[ #]] && IntegerQ[Sqrt[IntegerReverse[#]]]&] (* Harvey P. Dale, Jul 19 2023 *)

Formula

a(n) = A035123(n)^2. - R. J. Mathar, Jan 25 2017

A061457 Squares whose reversal is also a square.

Original entry on oeis.org

0, 1, 4, 9, 100, 121, 144, 169, 400, 441, 484, 676, 900, 961, 1089, 9801, 10000, 10201, 10404, 10609, 12100, 12321, 12544, 12769, 14400, 14641, 14884, 16900, 40000, 40401, 40804, 44100, 44521, 44944, 48400, 48841, 67600, 69696, 90000, 90601
Offset: 1

Views

Author

Amarnath Murthy, May 03 2001

Keywords

Comments

The corresponding square roots are in A102859.

Examples

			169 and 961 are both squares.
1089 = 33^2 and 9801 = 99^2 so 1089 and 9801 belong to the sequence.

Links

Crossrefs

Cf. A102859 (square roots), A033294 (exclude final digit 0).

Programs

  • Magma
    [n^2: n in [0..306] | IsSquare(Seqint(Reverse(Intseq(n^2))))]; // Bruno Berselli, Apr 30 2011
  • Mathematica
    Select[Range[0,400]^2,IntegerQ[Sqrt[IntegerReverse[#]]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 29 2019 *)
  • PARI
    { for(m=0, 1000, my(r=fromdigits(Vecrev(digits(m^2)))); if(issquare(r), print1(m^2, ", ") )) } \\ Harry J. Smith, Jul 23 2009
  • Python
    from itertools import count, islice
from sympy import integer_nthroot
def A061457_gen(): # generator of terms
    return filter(lambda n:integer_nthroot(int(str(n)[::-1]),2)[1], (n**2 for n in count(0)))
A061457_list = list(islice(A061457_gen(),30)) # Chai Wah Wu, Nov 18 2022

Formula

a(n) = A102859(n)^2.

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), May 17 2001

A064021 Squares k^2 such that reverse(k)^2 = reverse(k^2), excluding squares of palindromes.

Original entry on oeis.org

144, 169, 441, 961, 10404, 10609, 12544, 12769, 14884, 40401, 44521, 48841, 90601, 96721, 1004004, 1006009, 1022121, 1024144, 1026169, 1042441, 1044484, 1062961, 1212201, 1214404, 1216609, 1236544, 1238769, 1256641, 1258884, 1442401
Offset: 1

Views

Author

Harvey P. Dale, Sep 18 2001

Keywords

Comments

Subsequence of A035090. - M. F. Hasler, Mar 22 2011

Examples

			1026169 is included because its square root, 1013, when reversed (i.e., 3101) and squared yields 9616201.
Squares < 10 and 121 = 11^2, 484 = 22^2, ... are not in the sequence, since they are the square of a palindrome. - _M. F. Hasler_, Mar 22 2011

References

  • David Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 124, 127 (Rev. ed. 1997).

Links

Crossrefs

Cf. A035124, A035123, A033294, A140212.

Programs

  • Mathematica
    Cases[Range[2000]^2, k_ /; Mod[k, 10] != 0 && IntegerDigits[k] != Reverse[IntegerDigits[k]] && FromDigits[Reverse[IntegerDigits[Sqrt[k]]]]^2 == FromDigits[Reverse[IntegerDigits[k]]]] (* Jean-François Alcover, Mar 22 2011 *)
Select[Range[1250]^2,!PalindromeQ[Sqrt[#]]&&IntegerReverse[#] == IntegerReverse[ Sqrt[#]]^2 &&Mod[#,10]!=0&] (* Harvey P. Dale, Jul 01 2022 *)
  • PARI
    Rev(x)= { local(d,r); r=0; while (x>0, d=x-10*(x\10); x\=10; r=r*10 + d); return(r) }
{ n=0; for (m=1, 10^9, if (m%10==0, next); x=m^2; r=Rev(x); if (r==x, next); if (r==Rev(m)^2, write("b064021.txt", n++, " ", x); if (n==750, break)) ) } \\ Harry J. Smith, Sep 06 2009

Formula

{n = A000290(k) such that A004086(A000290(k)) = A000290(A004086(k)) and k is not in A002113}. - Jonathan Vos Post, May 02 2011
a(n) = A140212(n)^2. - Giovanni Resta, Jun 22 2018

A115690 Squares whose digit reversal is a powerful(1) number (A001694).

Original entry on oeis.org

1, 4, 9, 100, 121, 144, 169, 400, 441, 484, 576, 676, 900, 961, 1089, 9801, 10000, 10201, 10404, 10609, 12100, 12321, 12544, 12769, 14400, 14641, 14884, 16900, 25281, 27225, 40000, 40401, 40804, 44100, 44521, 44944, 48400, 48841, 57600
Offset: 1

Views

Author

Giovanni Resta, Jan 31 2006

Keywords

Comments

If x is a member, then so is 100*x. - Robert Israel, Mar 16 2020

Examples

			25281=159^2 and 18252=2^2*3^3*13^2 is powerful.

Links

Crossrefs

Subsequence of A115656.
A033294 is a subsequence, and the main entry for this sequence.
Cf. A001694, A115689.

Programs

  • Maple
    filter:= proc(n) local L,i,x;
  L:= convert(n,base,10);
  x:=add(L[-i]*10^(i-1),i=1..nops(L));
  andmap(t -> t[2]>=2, ifactors(x)[2]):
end proc:select(filter, [seq(i^2,i=1..10^4)]); # Robert Israel, Mar 16 2020
  • PARI
    is(k) = ispowerful(fromdigits(Vecrev(digits(k))));
select(is, vector(300, n, n^2)) \\ Michel Marcus, Nov 01 2022

Formula

Trivially, n^2 <= a(n) <= 100^(n-1). - Charles R Greathouse IV, Nov 01 2022

A354256 Squares that remain square when written backward, are not divisible by 10, and have an even number of digits.

Original entry on oeis.org

1089, 9801, 698896, 10036224, 42263001, 637832238736, 1021178969603881, 1883069698711201, 4099923883299904, 6916103777337773016196
Offset: 1

Views

Author

Jon E. Schoenfield, May 21 2022

Keywords

Comments

a(10) > 10^21.
Is this sequence infinite?
Every term is a multiple of 121.
Terms come in nonpalindromic pairs and palindromic singles; see Example section.
Removal of the "even number of digits" requirement yields A033294, which has 8560 terms < 10^20.
A027829 is a subsequence. - Chai Wah Wu, May 23 2022

Examples

			There are no 2-digit terms.
The smallest 4-digit multiple of 121 is 1089 = 33^2, which happens to be a(1); its digit reversal is a(2) = 9801 = 99^2.
The only 6-digit term is the palindrome a(3) = 698896 = 836^2.
The only 8-digit terms are a(4) = 10036224 = 3168^2 and its digit reversal a(5) = 42263001 = 6501^2.
There are no 10-digit terms.
The only 12-digit term is the palindrome a(6) = 637832238736 = 798644^2.
There are no 14-digit terms.
There are three 16-digit terms: a(7) = 1021178969603881 = 31955891^2, its digit reversal a(8) = 1883069698711201 = 43394351^2, and the palindrome a(9) = 4099923883299904 = 64030648^2.

Links

Crossrefs

Cf. A027829, A033294.

Programs

  • Mathematica
    Select[Range[500000]^2,EvenQ[IntegerLength[#]]&&Mod[#,10]!=0&&IntegerQ[Sqrt[ IntegerReverse[ #]]]&] (* The program generates the first five terms of the sequence. *) (* Harvey P. Dale, Jul 28 2024 *)
  • Python
    from math import isqrt
from itertools import count, islice
def sqr(n): return isqrt(n)**2 == n
def agen(): yield from (k*k for k in count(1) if k%10 and len(s:=str(k*k))%2==0 and sqr(int(s[::-1])))
print(list(islice(agen(), 6))) # Michael S. Branicky, May 23 2022
  • Python
    from math import isqrt
from itertools import count, islice
from sympy import integer_nthroot
def A354256_gen(): # generator of terms
    for l in count(2,2):
        for m in (1,4,5,6,9):
            for k in range(1+isqrt(m*10**(l-1)-1),1+isqrt((m+1)*10**(l-1)-1)):
                if k % 10 and integer_nthroot(int(str(k*k)[::-1]),2)[1]:
                    yield k*k
A354256_list = list(islice(A354256_gen(),9)) # Chai Wah Wu, May 23 2022

Extensions

a(10) from Chai Wah Wu, May 24 2022

A035122 Roots of 'squares remaining square when written backwards'.

Original entry on oeis.org

1, 2, 3, 11, 12, 13, 21, 22, 26, 31, 33, 99, 101, 102, 103, 111, 112, 113, 121, 122, 201, 202, 211, 212, 221, 264, 301, 307, 311, 836, 1001, 1002, 1003, 1011, 1012, 1013, 1021, 1022, 1031, 1101, 1102, 1103, 1111, 1112, 1113, 1121, 1122, 1201, 1202, 1211
Offset: 1

Views

Author

Patrick De Geest, Nov 15 1998

Keywords

Comments

Numbers with trailing zeros are excluded.

Examples

			99^2 = 9801 -> 1089 = 33^2.

Links

Crossrefs

Cf. A033294, A035125.

Programs

  • Mathematica
    Select[Range[10000], IntegerQ[Sqrt[FromDigits[Reverse[IntegerDigits[ #^2]]]]] &] (* and then delete terms ending with 0 - N. J. A. Sloane, Jul 08 2011 *)
Sqrt[Select[Range[2000]^2, Mod[#, 10]!=0&&IntegerQ[Sqrt[FromDigits[Reverse[IntegerDigits[#]]]]]&]] (* Vincenzo Librandi, Sep 22 2015 *)

Formula

a(n) = sqrt(A033294(n)). - Michel Marcus, Sep 22 2015
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.