[ January 15, 2015 ]
n and n_reversed are square numbers
whose roots may also be [ pseudo-] reversible
Andres Molina from Tulsa (email)

1089 is a smaller square number example whereby its reverse is also a square number
(n being not palindromic and without ending with zero).

12303690084 is the smallest square number whose reversal is also a square number
but so that their roots are pseudo-reversals amongst themselves (in base 10 at least).

What are the methods to transform nonreversible pairs into genuine reversibles ?
One way is to replace substrings using value n = – 1 like follows :

09 (leading zero included) can be replaced by 1n : which is in fact 10 – 1
099 (leading zero included) can be replaced by 10n : which is in fact 100 – 1
19 can be replaced by 2n : which is in fact 20 – 1
199 can be replaced by 20n : which is in fact 200 – 1
etc.

Let me clarify with next pair of reversible squares 12303690084 & 48009630321
110922 = 111n22
and
219111 = 22n111
As you can see now the roots 111n22 and 22n111 have changed into reversibles.

Another larger example with 102030003060900000804 & 408000009060300030201
10100990202 = 101010n0202
and
20199010101 = 2020n010101
And hey presto the reversibles 101010n0202 and 2020n010101 pop up.

Can you find more such number pairs whose square roots are [pseudo-] reversals as well ?
Or larger sporadic numbers beyond those listed in A061457 and A156316 ?

When n and R(n) are equal they are of course palindromic.
Please link to Palindromic Squares.

Square Root n	(n)	Reversal(n)	Square Root R(n) [Pseudo]Reversal of Square Root n
122	144	441	212
132	169	961	312
332	1089	9801	992
1022	10404	40401	2012
1032	10609	90601	3012
GAP
31682	10036224	42263001	65012
205082	420578064	460875024	214682
1109222 = 111n222	12303690084	48009630321	2191112 = 22n1112
3035772	92158994929	92949985129	3048772
11009222 = 1101n222	1212029250084	4800529202121	21910112 = 22n10112
11092112 = 111n2112	1230349042521	1252409430321 °° root is palindromic !	11191112 °° = 112n1112
11109222 = 1111n222	1234147690084	4800967414321	21911112 = 22n11112
30803672	9488660854689	9864580668849	31407932
101109222 = 10111n222	102230743690084	480096347032201	219111012 = 22n111012
110091112 = 1101n1112	121200525010321	123010525002121	110910112 = 111n10112
110091222 = 1101n1222	121200767210884	488012767002121	220910112 = 221n10112
110092212 = 1101n2212	121202947026841	148620749202121	121910112 = 122n10112
110191112 = 1102n1112	121420807230321	123032708024121	110920112 = 111n20112
110910222 = 111n10222	123010769004484	484400967010321	220091112 = 2201n1112
110911112 = 111n11112	123012743214321	123412347210321	111091112 = 1111n1112
110911212 = 111n11212	123012965036641	146630569210321	121091112 = 1211n1112
110911222 = 111n11222	123012987218884	488812789210321	221091112 = 2211n1112
110912022 = 111n12022	123014761804804	408408167410321	202091112 = 2021n1112
110912112 = 111n12112	123014961446521	125644169410321	112091112 = 1121n1112
110912122 = 111n12122	123014983628944	449826389410321	212091112 = 2121n1112
110921112 = 111n21112	123034926436321	123634629430321	111191112 = 1112n1112
111009222 = 11101n222	123230469250084	480052964032321	219101112 = 22n101112
111092112 = 1111n2112	123414569042521	125240965414321	111911112 = 112n11112
111109222 = 11111n222	123452587690084	480096785254321	219111112 = 22n111112
260490132	678551078274169	961472870155876	310076262
319558912	1021178969603881	1883069698711201	433943512
101009902022 = 101010n02022	102030003060900000804	408000009060300030201	201990101012 = 2020n0101012
Related OEIS sequence A115656 - Both n and the reverse of n are powerful(1) numbers (A001694). A061457 - Numbers n such that n and its reversal are both squares. A156316 - Perfect squares with property that their digit reversal is a larger perfect square. A129914 - Irregular square reversible numbers. Numbers which when squared and written backwards give a square again, but don't satisfy reverse(n^2)=reverse(n)^2. Index entry for sequences related to reversing digits of squares