A083408 -id:A083408 - cp's OEIS frontend

A117281 Palindromes whose squares belong to A066531.

252, 2772, 20502, 23632, 25452, 26962, 27972, 48384, 225522, 252252, 259952, 279972, 619916, 1113111, 1226221, 1357531, 2005002, 2070702, 2126212, 2150512, 2216122, 2226222, 2249422, 2275722, 2316132, 2347432, 2386832, 2429242

Offset: 1

Lekraj Beedassy, Apr 23 2006

a(n)^2 = A083408(k) for some k.

			27972 is in the sequence because 782432784 = 27972*27972 = 15984*48951.
1113111 is in the sequence because 1239016098321 = 1113111*1113111 = 1022121*1212201.

Chai Wah Wu, Table of n, a(n) for n = 1..4679 (n = 1..54 from Hans Havermann)
Shyam Sunder Gupta, EPRNs

Cf. A066531, A083408 (square EPRNs), A206642.

Edited and corrected by Klaus Brockhaus, Aug 21 2007

Edited by N. J. A. Sloane, Aug 01 2019

A325148 Squares which can be expressed as the product of a number and its reversal.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 400, 484, 900, 1089, 1600, 1936, 2500, 3025, 3600, 4356, 4900, 5929, 6400, 7744, 8100, 9801, 10000, 10201, 12100, 12321, 14641, 17161, 19881, 22801, 25921, 29241, 32761, 36481, 40000, 40804, 44944, 48400, 49284, 53824, 58564, 63504, 68644, 73984, 79524, 85264

Offset: 1

Bernard Schott, Apr 03 2019

The numbers k such that k * rev(k) is a square are in A306273.
The squares of palindromes of A014186 are a subsequence.
The square roots of the first 65 terms of this sequence (from 0 to 160000) are exactly the first 65 terms of A061917. Then a(66) = 162409 = 403^2 and the non-palindrome 403 is the first term of another sequence A325151.

			Zero ways: 169 = 13^2 cannot be equal to k * rev(k).
One way: 400 = 200 * 2; 10201 = 101 * 101; 162409 = 169 * 961.
Two ways: 7683984 = 2772 * 2772 = 1584 * 4851.
Three ways: 6350400 = 14400 * 441 = 25200 * 252 = 44100 * 144.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms for n = 1..231 from R. J. Mathar)
Bernard Schott, The different ways

Equals A325149 Union A083408.
Cf. A325149 (only one way), A083408 (at least two ways). A325150 (exactly two ways), A307019 (exactly three ways).
Subsequences: A014186 (square of palindromes), A076750 (product of a non-palindrome and its reversal, where leading zeros are not allowed).
Cf. A061917, A325151 (some square roots of this sequence).

isA305231 := proc(n)
    local d;
    for d in numtheory[divisors](n) do
        if d = digrev(n/d) then
            return true ;
        end if;
    end do:
    false ;
end proc:
n := 1;
for i from 0 to 4000 do
    i2 := i^2 ;
    if isA305231(i2) then
        printf("%d %d\n",n,i2) ;
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Aug 09 2019

{0}~Join~Select[Range[10^3]^2,(d1=Select[Divisors[n=#],#<=Sqrt@n&];Or@@Table[d1[[k]]==(IntegerReverse/@(n/d1))[[k]],{k,Length@d1}])&] (* Giorgos Kalogeropoulos, Jun 09 2021 *)

from sympy import divisors
A325148_list = [0]
for n in range(10**6):
    n2 = n**2
    for m in divisors(n2):
        if m > n:
            break
        if m == int(str(n2//m)[::-1]):
            A325148_list.append(n2)
            break # Chai Wah Wu, Jun 09 2021

Intersection of A305231 and A000290. - R. J. Mathar, Aug 09 2019

Definition corrected by N. J. A. Sloane, Aug 01 2019

A325150 Squares which can be expressed as the product of a number and its reversal in exactly two ways.

Original entry on oeis.org

63504, 435600, 7683984, 16240900, 25401600, 66585600, 420332004, 558471424, 647804304, 726949444, 782432784, 1067328900, 1624090000, 1897473600, 2341011456, 2540160000, 6658560000, 50860172484, 52587662400, 63631071504, 67575042304, 78384320784, 96118600900, 106732890000

Offset: 1

Bernard Schott, Apr 03 2019

When q = m^2 does not end with a 0 is a term, then m is a palindrome belonging to A117281.

When q = m^2 ending with a 0 is a term, then either m = r * 10^u where r belongs to A325151 and u >= 1, or m is in A342994.

			1) Squares without trailing zeros:
Even square: 7683984 = 2772^2 = 2772 * 2772 = 1584 * 4851.
Odd square: 1239016098321 = 1113111^2 = 1113111 * 1113111 = 1022121 * 1212201.
2) Squares with trailing zeros:
1st case: 16240900 = 4030^2 = 16900 * 961 = 96100 * 169.
2nd case: 435600 = 660^2 = 6600 * 66 = 528 * 825.

D. Wells, 63504 entry, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1997, p. 168.

Chai Wah Wu, Table of n, a(n) for n = 1..1672
David A. Corneth, Examples of factorizations of terms in exactly two or three ways.
Shyam Sunder Gupta, EPRN Numbers.

Cf. A325148 (at least one way), A325149 (only one way), A083408 (at least two ways), A307019 (exactly three ways).
Cf. A083407 (odd squares), A083408 (even squares without trailing 0's).
Cf. A117281, A325151, A342994.

Corrected terms by Chai Wah Wu, Apr 12 2019

Definition corrected by N. J. A. Sloane, Aug 01 2019

A307019 Squares which can be expressed as the product of a number and its reversal in exactly three different ways.

Original entry on oeis.org

6350400, 43560000, 635040000, 768398400, 4356000000, 42033200400, 55847142400, 63504000000, 64780430400, 72694944400, 76839840000, 78243278400, 234101145600, 435600000000, 4203320040000, 5086017248400, 5584714240000, 6350400000000, 6363107150400, 6478043040000, 6757504230400

Offset: 1

Bernard Schott, Mar 20 2019

1) Why do all these terms end with an even number of zeros?
1.1) Is it possible to find a term that does not end with zeros? If such a term m exists, this number must satisfy the Diophantine equation m^2 = a*rev(a) = b*rev(b) = c*rev(c). No solution (m,a,b,c) with m that does not end with zeros is known.
1.2) Consider now the Diophantine equation: m^2 = a*rev(a) = b*rev(b) where a is a palindrome and b is not a palindrome. For each solution (m,a,b), we generate terms (10*m)^2 of this sequence and we get: (10*m)^2 = 100 * m^2 = (100*a)*(rev(100*a) = (100*b)*(rev(100*b)) = (100*rev(b)) * (rev(100*rev(b))).
Example: with a(1) = 63504 = 252^2 = 252 * 252 = 144 * 441, so (m,a,b) = (63504,252,144), we obtain the 3 following ways: 6350400 = 25200 * 252 = 14400 * 441 = 44100 * 144.
2) When can square numbers be expressed in this way in more than three different ways?
If the Diophantine equation: m^2 = a*rev(a) = b*rev(b), with a <> b and a and b not palindromes has a solution, then it is possible to get integers equal to (10*m)^2 which can be expressed as the product of a number and its reversal in exactly four different ways.
We don't know if such a solution (m,a,b) exists.
David A. Corneth has found 70 terms < 6*10^15 belonging to this sequence (see links in A083408), but no square has four solutions for m^2 = k * rev(k) until 6*10^15.
There is no square less than 10^24 with 4 or more different ways. - Chai Wah Wu, Apr 12 2019

			6350400 = 2520^2 = 25200 * 252 = 14400 * 441 = 44100 * 144.
43560000 = 6600^2 = 660000 * 66 = 52800 * 825 = 82500 * 528.

Chai Wah Wu, Table of n, a(n) for n = 1..826 (n = 1..70 from David A. Corneth)
David A. Corneth, Examples of factorizations of terms as described in Name.
Shyam Sunder Gupta, Equal Product Of Reversible Numbers

Subsequence of A083406 and A083408.

Cf. A002113, A004086, A306273, A322835.

is(n) = {if(!issquare(n), return(0)); my(d = divisors(n), t = 0); forstep(i = #d, #d \ 2 + 1, -1, revd = fromdigits(Vecrev(digits(d[i]))); if(revd * d[i] == n, t++; if(t > 3, return(0)); ) ); t==3 } \\ David A. Corneth, Mar 20 2019

Corrected and extended by David A. Corneth, Mar 20 2019

Definition corrected and entry edited by N. J. A. Sloane, Aug 01 2019

A325149 Squares which can be expressed as the product of a number and its reverse in exactly one way.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 400, 484, 900, 1089, 1600, 1936, 2500, 3025, 3600, 4356, 4900, 5929, 6400, 7744, 8100, 9801, 10000, 10201, 12100, 12321, 14641, 17161, 19881, 22801, 25921, 29241, 32761, 36481, 40000, 40804, 44944, 48400, 49284, 53824, 58564, 68644, 73984, 79524, 85264, 90000

Offset: 1

Bernard Schott, Apr 03 2019

The first 47 terms of this sequence (from 0 to 58564) are identical to the first 47 terms of A325148. The square 63504 is not present because it can be expressed in two ways: 63504 = 252 * 252 = 144 * 441.
There are three families of squares in this sequence:
1) Squares of palindromes in A002113\A117281.
2) Squares of non-palindromes which form the sequence A325151.
These squares are a subsequence of A076750.
3) Squares of (m*10^q) with q >= 1 and m palindrome in A002113\A117281.

			For each family:
1) Square of palindromes: 53824 = 232^2 = 232 * 232.
2) Square of non-palindromes m^2 = k*rev(k) with k and rev(k) which have the same number of digits: 162409 = 403^2 = 169 * 961.
3) Square ends with zeros: 48400 = 220^2 = 2200 * 22.

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition, p. 168.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A325148 (at least one way), A083408 (at least two ways), A325150 (exactly two ways), A307019 (exactly three ways).

Cf. A014186 (squares of palindromes), A076750.

a(52) corrected by Chai Wah Wu, Apr 11 2019

Definition corrected by N. J. A. Sloane, Aug 01 2019

A083406 Even squares which can be expressed as the product of a number and its reversal in at least two different ways.

Original entry on oeis.org

63504, 435600, 6350400, 7683984, 16240900, 25401600, 43560000, 66585600, 420332004, 558471424, 635040000, 647804304, 726949444, 768398400, 782432784, 1067328900, 1624090000, 1897473600, 2341011456, 2540160000

Offset: 1

Shyam Sunder Gupta, Jun 07 2003

For n=1..49 identical to A083408.

			63504 = 252 * 252 = 144 * 441, 7683984 = 2772 * 2772 = 1584 * 4851, etc.

S. S. Gupta, EPRNs, Science Today, Feb. 1987, India.

Chai Wah Wu, Table of n, a(n) for n = 1..2372 (first 73 terms from Hans Havermann)
Shyam Sunder Gupta, Equal Product of Reversible Numbers

Cf. A031877, A066531, A083407 (odd squares version), A083408 (all squares version).

Corrected and extended by Hans Havermann, Feb 11 2012

Definition corrected by N. J. A. Sloane, Aug 01 2019

A083407 Odd squares which can be expressed as the product of a number and its reversal in at least two different ways.

Original entry on oeis.org

1239016098321, 1503617940841, 1842890415961, 11151144884889, 12909311260209, 149920947896841, 181937770841761, 12128839583882121, 12598313930168521, 12639203218972521, 13081277143774921, 14462953695004441, 14999934809556841, 15260975534573041

Offset: 1

Shyam Sunder Gupta, Jun 07 2003

			1239016098321 = 1113111 * 1113111 = 1022121 * 1212201, etc.

S. S. Gupta, EPRNs, Science Today, Feb. 1987, India.

Chai Wah Wu, Table of n, a(n) for n = 1..126
Shyam Sunder Gupta, EPRN Numbers.

Cf. A066531, A083406 (even squares version), A083408 (all squares version).

Corrected by Hans Havermann, Feb 13 2012
a(4)-a(14) (including four found by Hans Havermann) from Donovan Johnson, Feb 18 2012
Definition corrected by N. J. A. Sloane, Aug 01 2019

A306273 Numbers k such that k * rev(k) is a square, where rev=A004086, decimal reversal.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 100, 101, 111, 121, 131, 141, 144, 151, 161, 169, 171, 181, 191, 200, 202, 212, 222, 232, 242, 252, 262, 272, 282, 288, 292, 300, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 400, 404, 414, 424, 434, 441, 444, 454, 464, 474, 484, 494, 500, 505, 515, 525, 528, 535

Offset: 1

Bernard Schott, Feb 02 2019

The first nineteen terms are palindromes (cf. A002113). There are exactly seven different families of integers which together partition the terms of this sequence. See the file "Sequences and families" for more details, comments, formulas and examples.
From Chai Wah Wu, Feb 18 2019: (Start)
If w is a term with decimal representation a, then the number n corresponding to the string axa is also a term, where x is a string of k repeated digits 0 where k >= 0. The number n = w*10^(k+m)+w = w*(10^(k+m)+1) where m is the number of digits of w. Then R(n) = R(w)*10^(k+m)+R(w) = R(w)(10^(k+m)+1). Then n*R(n) = w*R(w)(10^(k+m)+1)^2 which is a square since w is a term.
The same argument shows that numbers corresponding to axaxa, axaxaxa, ... are also terms.
For example, since 528 is a term, so are 528528, 5280528, 52800528, 5280052800528, etc.
(End)

			One example for each family:
family 1 is A002113: 323 * 323 = 323^2;
family 2 is A035090: 169 * 961 = 13^2 * 31^2 = 403^2;
family 3 is A082994: 288 * 882 = (2*144) * (2*441) = 504^2;
family 4 is A002113(j) * 100^k: 75700 * 757 = 7570^2;
family 5 is A035090(j) * 100^k: 44100 * 144 = 2520^2;
family 6 is A082994(j) * 100^k: 8670000 * 768 = 81600^2;
family 7 is A323061(j) * 10^(2k+1): 5476580 * 856745 = 2166110^2.

C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, NY. (1966), pp. 88-89.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, Revised edition (1997), p. 168.

Robert Israel, Table of n, a(n) for n = 1..10000
Bernard Schott, Sequences and Families
Eric Weisstein's World of Mathematics, Reversal

Cf. A002113, A070760, A062917, A035090, A082994, A322835, A323061.

Cf. A083406, A083407, A083408, A117281 (Squares = k * rev(k) in at least two ways).

revdigs:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
filter:= n -> issqr(n*revdigs(n)):
select(filter, [$0..1000]);# Robert Israel, Feb 09 2019

Select[Range[0, 535], IntegerQ@ Sqrt[# IntegerReverse@ #] &] (* Michael De Vlieger, Feb 03 2019 *)

isok(n) = issquare(n*fromdigits(Vecrev(digits(n)))); \\ Michel Marcus, Feb 04 2019

A206642 Non-palindromes whose squares are in A066531.

Original entry on oeis.org

660, 2520, 4030, 5040, 6600, 8160, 25200, 27720, 32670, 40300, 43560, 50400, 66000, 81600, 205020, 229320, 236320, 254520, 269620, 277200, 279720, 310030, 326700, 329670, 351430, 410040, 435600, 439560, 458640, 483840, 486160

Offset: 1

Hans Havermann, Feb 11 2012

			660 is in the sequence because 660^2 = 528*825 = 6600*0066; 2520 is in the sequence because 2520^2 = 14400*00441 = 25200*00252 = 44100*00144; etc.

Shyam Sunder Gupta, EPRNs

Cf. A066531, A083408 (square EPRNs), A117281 (palindromic version).

Edited by N. J. A. Sloane, Aug 01 2019

A325152 Numbers whose squares can be expressed as the product of a number and its reversal.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 22, 30, 33, 40, 44, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 101, 110, 111, 121, 131, 141, 151, 161, 171, 181, 191, 200, 202, 212, 220, 222, 232, 242, 252, 262, 272, 282, 292, 300, 303, 313, 323, 330, 333, 343, 353, 363, 373, 383, 393, 400, 403, 404, 414, 424, 434

Offset: 1

Bernard Schott, Apr 11 2019

The corresponding squares are in A325148 and the numbers k such that k * rev(k) is a square are in A306273.
The squares of the first 47 terms of this sequence (from 0 to 242) can be expressed as the product of a number and its reversal in only one way; then a(48) = 252 and 252^2 = 252 * 252 = 144 * 441.
The first 65 terms of this sequence (from 0 to 400) are exactly the first 65 terms of A061917; then a(66) = 403, non-palindrome, is the first term of the sequence A325151.

			One way: 20^2 = 400 = 200 * 2.
Two ways: 2772^2 = 7683984 = 2772 * 2772 = 1584 * 4851.
Three ways: 2520^2 = 14400 * 441 = 25200 * 252 = 44100 * 144.
403 is a member since 403^2 = 162409 = 169*961 (note that 403 is not a member of A281625).

Cf. A325148, A325149, A083408, A325150, A307019.
Cf. also A061917, A325151.
Similar to but different from A281625.

a(n) = sqrt(A325148(n)).

cp's OEIS Frontend

A117281 Palindromes whose squares belong to A066531.

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A325148 Squares which can be expressed as the product of a number and its reversal.

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A325150 Squares which can be expressed as the product of a number and its reversal in exactly two ways.

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A307019 Squares which can be expressed as the product of a number and its reversal in exactly three different ways.

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A325149 Squares which can be expressed as the product of a number and its reverse in exactly one way.

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A083406 Even squares which can be expressed as the product of a number and its reversal in at least two different ways.

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A083407 Odd squares which can be expressed as the product of a number and its reversal in at least two different ways.

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A306273 Numbers k such that k * rev(k) is a square, where rev=A004086, decimal reversal.

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