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

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

1,3

Comments

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.

References

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

Links

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

Example

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.

Crossrefs

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.

Sequence in context: A025741 A179459 A325148 * A326707 A352618 A030476

Adjacent sequences: A325146 A325147 A325148 * A325150 A325151 A325152

Keyword

nonn,base

Author

Bernard Schott, Apr 03 2019

Extensions

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

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

Status

approved