A066531 -id:A066531 - cp's OEIS frontend

A066590 Numbers which can be expressed as product of a number and its reversal in at least three different ways.

635040, 1015560, 1446480, 1854720, 4356000, 6350400, 10155600, 14464800, 18547200, 43560000, 51665040, 59552640, 63504000, 67944240, 76839840, 78127560, 82623060, 88560360, 95236960, 99497160, 101556000, 105094080

Offset: 1

Robert G. Wilson v, Jan 08 2002

nonn

A066598 is a subsequence, e.g., a(2)=A066598(1). - M. F. Hasler, Feb 14 2012

			63540 = 1440 * 411 = 2520 * 252 = 4410 * 144.

Shyam Sunder Gupta, Equal Product of Reversible Numbers (EPRN), Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 12, 353-365.

Cf. A066531, A066598.

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]] - 2, a[[ # ]] == a[[ # + 2 ]] & ]]]

Definition clarified by M. F. Hasler, Feb 14 2012

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

A066599 a(n) = smallest number which can be expressed as the product of a number and its reversal in exactly n different ways.

Original entry on oeis.org

1, 2520, 635040, 1015560, 10119126106147652568, 11158922880, 5918858243794044864, 1046458990080

Offset: 1

Robert G. Wilson v, Jan 08 2002

nonn

a(6) = 11158922880 and a(8) = 1046458990080.

a(10) = 101191261061476525680, a(12) = 62624600899319949840, a(14) = 59188582437940448640. - Chai Wah Wu, Apr 16 2019

			a(1) = 1 = 1*1; a(2) = 2520 = 120*21 = 210*12; a(3) = 635040 = 1440*441 = 2520*252 = 4410*144; a(4) = 1015560 = 1560*651 = 2730*372 = 3720*273 = 6510*156; a(6) = 11158922880 = 132480*84231 = 231840*48132 = 275040*40572 = 405720*27504 = 481320*23184 = 842310*13248; a(8) = 1236480*846321 = 2163840*483612 = 2329440*449232 = 2567040*407652 = 4076520*256704 = 4492320*232944 = 4836120*216384 = 8463210*123648.
From _Chai Wah Wu_, Apr 16 2019: (Start)
a(5) = 10119126106147652568 = 8848263411 * 1143628488 = 8044687521 * 1257864408 = 4884561702 * 2071654884 = 4440958722 * 2278590444 = 4082378742 * 2478732804
a(7) = 5918858243794044864 = 4834624221 * 1224264384 = 4439423331 * 1333249344 = 4036246641 * 1466426304 = 2762642412 * 2142462672 = 2566246032 * 2306426652 = 2536813332 * 2333186352 = 2511746532 * 2356471152
a(10) = 101191261061476525680 = 88482634110 * 1143628488 = 80446875210 * 1257864408 = 48845617020 * 2071654884 = 44409587220 * 2278590444 = 40823787420 * 2478732804 = 24787328040 * 4082378742 = 22785904440 * 4440958722 = 20716548840 * 4884561702 = 12578644080 * 8044687521 = 11436284880 * 8848263411
a(12) = 62624600899319949840 = 46891553310 * 1335519864 = 46055795310 * 1359755064 = 42632986410 * 1468923624 = 28946436120 * 2163464982 = 26795173320 * 2337159762 = 26317597320 * 2379571362 = 23795713620 * 2631759732 = 23371597620 * 2679517332 = 21634649820 * 2894643612 = 14689236240 * 4263298641 = 13597550640 * 4605579531 = 13355198640 * 4689155331
a(14) = 59188582437940448640 = 48346242210 * 1224264384 = 44394233310 * 1333249344 = 40362466410 * 1466426304 = 27626424120 * 2142462672 = 25662460320 * 2306426652 = 25368133320 * 2333186352 = 25117465320 * 2356471152 = 23564711520 * 2511746532 = 23331863520 * 2536813332 = 23064266520 * 2566246032 = 21424626720 * 2762642412 = 14664263040 * 4036246641 = 13332493440 * 4439423331 = 12242643840 * 4834624221
(End)

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, 10^7}]]; While[a[[1]] == 0, a = Drop[a, 1]]; Do[k = 1; While[ a[[k]] != a[[k + n - 1]], k++ ]; Print[ a[[k]]], {n, 1, 4} ]

a(5) and a(7) from Chai Wah Wu, Apr 16 2019

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

A346288 Base-10 numbers k such that k has no solutions to k = A * B and R(k) = R(A) * R(B) in any base 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

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17, 19, 20, 21, 23, 25, 29, 31, 32, 35, 37, 40, 41, 43, 47, 53, 55, 59, 60, 61, 65, 67, 70, 71, 73, 75, 77, 79, 80, 83, 85, 87, 89, 92, 94, 97, 98, 100, 101, 103, 106, 107, 109, 113, 114, 115, 127, 128, 129, 131, 137, 139, 141, 142, 145, 147

Offset: 1

Scott R. Shannon, Eric Angelini, and Carole Dubois, Jul 12 2021

This sequence uses the same rules to determine the numbers k as A346219 except that here the sequence includes only those numbers which have no solution to the two equalities k = A * B and R(k) = R(A) * R(B) in any base, from base 2 to base 10.

There are 8868747 terms less than 10 million. In that range the longest run where each consecutive number has one or more solutions to the equalities, thus do not appear in this sequence, is from 116 to 126.

			The first number not in the sequence is 14 as 14 = 2 * 7 and 22 = 2 * 11, and when written in base 5 those become 24 = 2 * 12 and 42 = 2 * 21. These satisfy the two equalities thus 14 is not a term in this sequence.
The second number not in the sequence is 16 as 16 = 2 * 8 and 26 = 2 * 13, which when written in base 6 become 24 = 2 * 12 and 42 = 2 * 21, which satisfy the equalities.
The third number not in the sequence is 18 as 18 = 2 * 9 and 30 = 2 * 15, which when written in base 7 become 24 = 2 * 12 and 42 = 2 * 21, which satisfy the equalities.

Cf. A346219, A346133, A066531, A230594, A001055, A038548.

A346290 Numbers k = s * t such that reverse(k) = reverse(s) * reverse(t) where reverse(k) is k with its digits reversed. A single-digit number is its own reversal and neither s nor t has a leading zero. No pair (s, t) has both s and t palindromic or single-digit.

Original entry on oeis.org

24, 26, 28, 36, 39, 42, 46, 48, 62, 63, 64, 68, 69, 82, 84, 86, 93, 96, 132, 143, 144, 154, 156, 165, 168, 169, 176, 187, 198, 204, 206, 208, 224, 226, 228, 231, 244, 246, 248, 252, 253, 264, 266, 268, 273, 275, 276, 284, 286, 288, 294, 297, 299, 306, 309

Offset: 1

Eric Angelini and Carole Dubois, Jul 13 2021

This sequence looks like A346133 but reversed products are here included.

			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.

Cf. A066531, A346133.

q[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, 300], q] (* Amiram Eldar, Jul 13 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 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(310)))) # Michael S. Branicky, Jul 13 2021

A386221 Numbers which can be expressed as the product of a number and its binary reversal in at least three different ways.

Original entry on oeis.org

2371610879733375, 8379443074856875, 103889625367330285, 162508095102648823, 2143169709271976875, 2481725627762299375, 4055619414785589625, 8167773178498814075, 9027536760163222895, 133527604616779133915, 133893081609954481115, 137216105281788994475, 457495296809227508125

Offset: 1

Zhao Hui Du, Aug 12 2025

It appears that most numbers that can be expressed in three different ways can also be expressed in four different ways. For a(n) < 2^88, only 11 numbers can be expressed in exactly three ways while 691 numbers can be expressed in exactly four ways.

			2371610879733375 = 51606261*45955875 = 64244529*36915375 = 64338225*36861615 while 51606261 = 11000100110111001011110101_2, 45955875 = 10101111010011101100100011_2 and reverse(11000100110111001011110101) = 10101111010011101100100011.
8379443074856875 = 101377465*82655875 = 102886105*81443875 = 114021425*73490075 = 115718225*72412475.

Shou'en Wang, Binary reversal (Chinese)

Cf. A066531, A066598.

cp's OEIS Frontend

A066590 Numbers which can be expressed as product of a number and its reversal in at least three different ways.

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A066599 a(n) = smallest number which can be expressed as the product of a number and its reversal in exactly n different ways.

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A346288 Base-10 numbers k such that k has no solutions to k = A * B and R(k) = R(A) * R(B) in any base 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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A346290 Numbers k = s * t such that reverse(k) = reverse(s) * reverse(t) where reverse(k) is k with its digits reversed. A single-digit number is its own reversal and neither s nor t has a leading zero. No pair (s, t) has both s and t palindromic or single-digit.

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A386221 Numbers which can be expressed as the product of a number and its binary reversal in at least three different ways.

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