A325148 -id:A325148 - cp's OEIS frontend

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

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

A076750 Squares which are the product of a non-palindrome and its reversal, where leading zeros are not allowed.

Original entry on oeis.org

63504, 162409, 254016, 435600, 665856, 7683984, 10673289, 18974736, 420332004, 525876624, 558471424, 647804304, 726949444, 782432784, 961186009, 1086823089, 1235030449, 1681328016, 1932129936, 2103506496, 2341011456, 2363515456, 3051678564, 3413130084, 4485784576

Offset: 1

Jason Earls, Nov 12 2002

			One way: 10673289 = 3267^2 = 1089*9801.
From _Bernard Schott_, Apr 12 2019: (Start)
Two ways:
  7683984 = 2772^2 = 2772*2772 = 1584*4851;
   435600 =  660^2 =  528*825  = 6600*66. (End)

Chai Wah Wu, Table of n, a(n) for n = 1..1169

Cf. A062917, A325149, A325150.

Subsequence of A325148.

More terms from Chai Wah Wu, Apr 11 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

A305231 Numbers that are the product of some integer and its digit reversal.

Original entry on oeis.org

0, 1, 4, 9, 10, 16, 25, 36, 40, 49, 64, 81, 90, 100, 121, 160, 250, 252, 360, 400, 403, 484, 490, 574, 640, 736, 765, 810, 900, 976, 1000, 1008, 1089, 1207, 1210, 1300, 1458, 1462, 1600, 1612, 1729, 1855, 1936, 1944, 2268, 2296, 2430, 2500, 2520, 2668, 2701

Offset: 1

Jon E. Schoenfield, May 27 2018

Terms of A061205, sorted in increasing order, with duplicates removed.

			12*21 = 252, so 252 is a term.
156*651 = 101556, so 101556 is a term. (It can also be written as 273*372; see A203924.)

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 1000 terms from Alois P. Heinz)

Cf. A061205, A203924.

Cf. A325148 (squares), A359981 (nonsquares).

a:= proc(n) option remember; local k, d; for k from 1+a(n-1) do
      for d in numtheory[divisors](k) do if k = d*(s-> parse(cat(
      seq(s[-i], i=1..length(s)))))(""||d) then return k fi od od
    end: a(1):=0:
seq(a(n), n=1..60);  # Alois P. Heinz, May 27 2018

a={0}; h=-1; For[k=0, k<=2701, k++, For[m=1, m<=DivisorSigma[0, k], m++, d=Divisors[k]; If[k/Part[d, m] == FromDigits[Reverse[IntegerDigits[Part[d, m]]]] && k>h , AppendTo[a, k]; h=k]]]; a (* Stefano Spezia, Jan 28 2023 *)

isok(n) = if (n==0, return (1), fordiv(n, d, if (n/d == fromdigits(Vecrev(digits(d))), return (1))); return (0)); \\ Michel Marcus, May 28 2018

A359981 Nonsquares which can be expressed as the product of a number and its digit reversal.

Original entry on oeis.org

10, 40, 90, 160, 250, 252, 360, 403, 490, 574, 640, 736, 765, 810, 976, 1000, 1008, 1207, 1210, 1300, 1458, 1462, 1612, 1729, 1855, 1944, 2268, 2296, 2430, 2520, 2668, 2701, 2944, 3154, 3478, 3627, 3640, 4000, 4030, 4032, 4275, 4606, 4840, 4930, 5092, 5605, 5740

Offset: 1

Stefano Spezia, Jan 20 2023

It contains all the numbers of the form i^2*10^(2*j+1).

			4840 = 220*22 = 22^2*10; 4930 = 58*85; 5092 = 67*76; 5605 = 59*95; 5740 = 140*41.

Cf. A000005, A000037, A000290, A004086, A027750, A305231 (supersequence).

Cf. A325148 (squares).

a={}; h=-1; For[k=0, k<=5750, k++, For[m=1, m<=DivisorSigma[0,k], m++, d=Divisors[k]; If[k/Part[d,m] == FromDigits[Reverse[IntegerDigits[Part[d,m]]]] && k>h && !IntegerQ[Sqrt[k]], AppendTo[a, k]; h=k]]]; a

Intersection of A305231 and A000037.

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)).

A345129 Sum of the squarefree products s*t from all positive integer pairs (s,t), such that s + t = n, s <= t.

Original entry on oeis.org

0, 1, 2, 3, 6, 5, 16, 22, 14, 21, 40, 46, 94, 46, 40, 109, 208, 159, 182, 161, 148, 268, 296, 380, 380, 472, 488, 497, 770, 620, 666, 851, 740, 1082, 560, 1015, 1506, 1226, 946, 1490, 2088, 1381, 2566, 1941, 2160, 2379, 2832, 2489, 2976, 3111, 2290, 3832, 4732, 3395, 3340

Offset: 1

Wesley Ivan Hurt, Jun 08 2021

nonn

			a(13) = 94; The partitions of 13 into two positive integer parts (s,t) where s <= t are (1,12), (2,11), (3,10), (4,9), (5,8), (6,7). The corresponding products are 1*12, 2*11, 3*10, 4*9, 5*8, and 6*7. The sum of the squarefree products from this list is 22 + 30 + 42 = 94.

Cf. A008683 (mu), A325148.

Table[Sum[k (n - k) MoebiusMu[k (n - k)]^2, {k, Floor[n/2]}], {n, 80}]

a(n) = Sum_{k=1..floor(n/2)} k * (n-k) * mu(k*(n-k))^2, where mu is the Möbius function (A008683).

cp's OEIS Frontend

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

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A076750 Squares which are the product of a non-palindrome and its reversal, where leading zeros are not allowed.

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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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A305231 Numbers that are the product of some integer and its digit reversal.

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A359981 Nonsquares which can be expressed as the product of a number and its digit reversal.

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

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A345129 Sum of the squarefree products s*t from all positive integer pairs (s,t), such that s + t = n, s <= t.

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