A066531 -id:A066531 - 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

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

A117282 Terms of A066531 not ending in 0.

Original entry on oeis.org

63504, 101556, 144648, 185472, 5166504, 5955264, 6794424, 7683984, 7812756, 8262306, 8856036, 9523696, 9949716, 10509408, 10865686, 11093796, 11768148, 11807208, 12494209, 13564768, 14201712, 15089472, 15449112, 16746912

Offset: 1

Lekraj Beedassy, Apr 23 2006

			63504 = 144*441 = 252*252; 7812756 = 1236*6321 = 2163*3612.

S. S. Gupta, "EPRNs" in Science Today (Subsequently renamed '2001'), pp. 77, Feb 1987, Times of India, Mumbai.

David W. Wilson, Hans Havermann, Table of n, a(n) for n = 1..10000 [ninety of David Wilson's terms had ended in zero; these were deleted and an additional ninety terms added by Hans Havermann, Feb 11 2012]
Shyam Sunder Gupta, EPRNs
Shyam Sunder Gupta, Unlucky 13, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 13, 353-365.

Cf. A066531 (EPRNs).

Edited and corrected by Klaus Brockhaus, Aug 21 2007

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

A083408 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, 4356000000, 6658560000, 42033200400, 50860172484, 52587662400

Offset: 1

Shyam Sunder Gupta, Jun 07 2003

Union of A083406 and A083407. - Lekraj Beedassy, Apr 23 2006

			63504 = 252 * 252 = 144 * 441,
1239016098321 = 1113111 * 1113111 = 1022121 * 1212201, etc.
635040000 = 144 * 4410000 = 252 * 2520000 = 441 * 1440000. - _David A. Corneth_, Mar 22 2019

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

Chai Wah Wu, Table of n, a(n) for n = 1..2498 (n = 1..76 from Hans Havermann, n = 77..241 from David A. Corneth)
David A. Corneth, Examples of factorizations of terms as described in Name.
Shyam Sunder Gupta, EPRN Numbers.
Shyam Sunder Gupta, Equal Product of Reversible Numbers (EPRN), Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 12, 353-365.

Cf. A062917, A066531, A083406 (even), A083407 (odd), A070760, A117281 (palindromic square roots), A206642 (non-palindromic square roots), A325150 (products in exactly two different ways), A307019 (products in exactly three different ways).

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 >= 2, return(1)); ) ); 0 } \\ David A. Corneth, Mar 21 2019

Corrected and extended by Hans Havermann, Feb 11 2012
a(21)-a(25) from David A. Corneth, Mar 21 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

A346133 Numbers N = A * B such that N (reversed digits) = A (reversed digits) * B (reversed digits). A single-digit number is its own reversal and neither A nor B has a leading zero. No pair (A, B) has both A and B palindromic or simple-digit. The reversed products are not included in the sequence.

Original entry on oeis.org

24, 26, 28, 36, 39, 46, 48, 68, 69, 132, 143, 144, 154, 156, 165, 168, 169, 176, 187, 198, 204, 206, 208, 224, 226, 228, 244, 246, 248, 252, 253, 264, 266, 268, 273, 275, 276, 284, 286, 288, 294, 297, 299, 306, 309, 336, 339, 366, 369, 374, 384, 385, 396, 399

Offset: 1

Eric Angelini and Carole Dubois, Jul 05 2021

			a(1) = 24 = 2 * 12 and 2 * 21 = 42 (which is 24 reversed);
a(2) = 26 = 2 * 13 and 2 * 31 = 62 (which is 26 reversed);
a(3) = 28 = 2 * 14 and 2 * 41 = 82 (which is 28 reversed);
a(4) = 36 = 3 * 12 and 3 * 21 = 63 (which is 36 reversed); etc.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A066531.

q[n_] := IntegerReverse[n] >= n && AnyTrue[Rest @ Take[(d = Divisors[n]), Ceiling[Length[d]/2]], (# > 9 || n/# > 9) && !Divisible[#, 10] && !Divisible[n/#, 10] && (!PalindromeQ[#] || !PalindromeQ[n/#]) && IntegerReverse[#] * IntegerReverse[n/#] == IntegerReverse[n] &]; Select[Range[2, 400], q] (* Amiram Eldar, Jul 07 2021 *)

from sympy import divisors
def rev(n): return int(str(n)[::-1])
def ok(n):
    divs = divisors(n)
    for a in divs[1:(len(divs)+1)//2]:
        b = n // a
        reva, revb, revn = rev(a), rev(b), rev(n)
        if revn < n or a%10 == 0 or b%10 == 0: continue
        if (reva != a or revb != b) and revn == reva * revb: return True
    return False
print(list(filter(ok, range(400)))) # Michael S. Branicky, Jul 06 2021

A066598 Numbers which can be expressed as the product of a number and its reversal in four different ways.

Original entry on oeis.org

1015560, 1446480, 1854720, 10155600, 14464800, 18547200, 51665040, 59552640, 67944240, 78127560, 82623060, 88560360, 95236960, 99497160, 101556000, 105094080, 108656860, 110937960, 117681480, 118072080, 124942090, 135647680

Offset: 1

Robert G. Wilson v, Jan 08 2002

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Shyam Sunder Gupta, Equal Product of Reversible Numbers (EPRN), Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 12, 353-365.

Cf. A066531.

f[n_] := (m = ToExpression[StringReverse[ToString[n]]]; If[n > m, n*m, 0]); a = Sort[ Table[ f[n], {n, 0, 10^6}]]; While[ a[[1]] == 0, a = Drop[a, 1]]; a[[ Select[ Range[ Length[a]] - 3, a[[ # ]] == a[[ # + 3 ]] & ]]]

A346219 Base-10 numbers k such that k can be written as k = A * B and R(k) = R(A) * R(B) in six or more bases, from base 2 to base 10, and where R(k), the digit reversal of k, is read as a number in the same base.

Original entry on oeis.org

1122, 17875, 65331, 367598, 818545, 1997905, 43998955, 100383283, 112887775, 112977865, 145683265, 230034805, 5231187650

Offset: 1

Scott R. Shannon, Eric Angelini and Carole Dubois, Jul 11 2021

This is a variation of the sequence A346133. Similar rules are used to determine the allowed values of A and B - neither number can have a leading 0, and both cannot be palindromes. However the reverse of k may appear as in general any solutions for k and R(k) will occur in different bases.
This sequence lists those base-10 numbers that meet these criteria in six or more bases, from base 2 to base 10. Note that, although k must stay the same when written in the different bases, the values of A and B need not be the same. Only the product of the chosen two factors and their reverses must equal k and R(k) in the given bases. See the example below and the linked data file.
No numbers are currently known that have solutions in seven or more bases. Assuming a(13) exists it is greater than 10^9.

			1122 is a term as k = A * B and R(k) = R(A) * R(B) has solutions in the six bases 4,5,7,8,9,10. See the table below for k = 1122.
.
      base   | k_base | A_base * B_base | R(k_base) | R(A_base) * R(B_base)
  =========================================================================
       4     | 101202 |    101 * 1002   |  202101   |       101 * 2001
  in base 10 |   1122 |     17 * 66     |    2193   |        17 * 129
  ------------------------------------------------------------------------
       5     |  13442 |      3 * 2444   |   24431   |         3 * 4442
  in base 10 |   1122 |      3 * 374    |    1866   |         3 * 622
  ------------------------------------------------------------------------
       7     |   3162 |     31 * 102    |    2613   |        13 * 201
  in base 10 |   1122 |     22 * 51     |     990   |        10 * 99
  -------------------------------------------------------------------------
       8     |   2142 |     21 * 102    |    2412   |        12 * 201
  in base 10 |   1122 |     17 * 66     |    1290   |        10 * 129
  -------------------------------------------------------------------------
       9     |   1476 |     12 * 123    |    6741   |        21 * 321
  in base 10 |   1122 |     11 * 102    |    4978   |        19 * 262
  -------------------------------------------------------------------------
      10     |   1122 |     11 * 102    |    2211   |        11 * 201
.
The bases used in the twelve terms below 10^9 are as follows:
.
         k    |       bases
  --------------------------------
        1122  |  4, 5, 7, 8, 9, 10
       17875  |  2, 3, 4, 6, 8, 10
       65331  |  2, 4, 5, 6, 8, 10
      367598  |  3, 4, 6, 8, 9, 10
      818545  |  2, 3, 4, 6, 8,  9
     1997905  |  2, 3, 4, 6, 8,  9
    43998955  |  2, 3, 4, 8, 9, 10
   100383283  |  2, 3, 4, 6, 9, 10
   112887775  |  2, 3, 4, 8, 9, 10
   112977865  |  2, 3, 4, 8, 9, 10
   145683265  |  2, 3, 4, 6, 8,  9
   230034805  |  2, 3, 4, 6, 8,  9
.

Scott R. Shannon, Factorizations of the twelve terms below 10^9.

Cf. A346133, A066531, A230594, A001055, A038548.

a(13) from Michael S. Branicky, Jun 21 2023

A083405 Odd numbers which can be expressed as the product of a number and its reversal in at least two different ways.

Original entry on oeis.org

12494209, 448583733, 494167063, 547084993, 851529915, 1191394449, 1213031209, 1338416599, 1343524977, 1464607989, 2064820905, 34509780333, 36738861453, 38016529663, 39118059873, 40127419683, 42087533593, 43093084603, 43754503723, 43948961613

Offset: 1

Shyam Sunder Gupta, Jun 07 2003

			12494209 = 13143 * 34131 = 31341 * 14313, 1191394449 = 12369 * 96321 = 30783 * 38707, etc.

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

Hans Havermann, Table of n, a(n) for n = 1..343
Shyam Sunder Gupta, EPRN Numbers.

Cf. A066531.

More terms from Hans Havermann, Feb 11 2012

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

cp's OEIS Frontend

A117281 Palindromes whose squares belong to A066531.

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A206642 Non-palindromes whose squares are in A066531.

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A117282 Terms of A066531 not ending in 0.

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A083408 Squares which can be expressed as the product of a number and its reversal in at least two different ways.

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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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Examples

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Links

Crossrefs

Extensions

A346133 Numbers N = A * B such that N (reversed digits) = A (reversed digits) * B (reversed digits). A single-digit number is its own reversal and neither A nor B has a leading zero. No pair (A, B) has both A and B palindromic or simple-digit. The reversed products are not included in the sequence.

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A066598 Numbers which can be expressed as the product of a number and its reversal in four different ways.

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Mathematica

A346219 Base-10 numbers k such that k can be written as k = A * B and R(k) = R(A) * R(B) in six or more bases, from base 2 to base 10, and where R(k), the digit reversal of k, is read as a number in the same base.

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