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%I #55 Feb 19 2019 04:47:02
%S 0,1,2,3,4,5,6,7,8,9,11,22,33,44,55,66,77,88,99,100,101,111,121,131,
%T 141,144,151,161,169,171,181,191,200,202,212,222,232,242,252,262,272,
%U 282,288,292,300,303,313,323,333,343,353,363,373,383,393,400,404,414,424,434,441,444,454,464,474,484,494,500,505,515,525,528,535
%N Numbers k such that k * rev(k) is a square, where rev=A004086, decimal reversal.
%C The first nineteen terms are palindromes (cf. A002113). There are exactly seven different families of integers which together partition the terms of this sequence. See the file "Sequences and families" for more details, comments, formulas and examples.
%C From _Chai Wah Wu_, Feb 18 2019: (Start)
%C If w is a term with decimal representation a, then the number n corresponding to the string axa is also a term, where x is a string of k repeated digits 0 where k >= 0. The number n = w*10^(k+m)+w = w*(10^(k+m)+1) where m is the number of digits of w. Then R(n) = R(w)*10^(k+m)+R(w) = R(w)(10^(k+m)+1). Then n*R(n) = w*R(w)(10^(k+m)+1)^2 which is a square since w is a term.
%C The same argument shows that numbers corresponding to axaxa, axaxaxa, ... are also terms.
%C For example, since 528 is a term, so are 528528, 5280528, 52800528, 5280052800528, etc.
%C (End)
%D C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, NY. (1966), pp. 88-89.
%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, Revised edition (1997), p. 168.
%H Robert Israel, <a href="/A306273/b306273.txt">Table of n, a(n) for n = 1..10000</a>
%H Bernard Schott, <a href="/A306273/a306273_2.pdf">Sequences and Families</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Reversal.html">Reversal</a>
%e One example for each family:
%e family 1 is A002113: 323 * 323 = 323^2;
%e family 2 is A035090: 169 * 961 = 13^2 * 31^2 = 403^2;
%e family 3 is A082994: 288 * 882 = (2*144) * (2*441) = 504^2;
%e family 4 is A002113(j) * 100^k: 75700 * 757 = 7570^2;
%e family 5 is A035090(j) * 100^k: 44100 * 144 = 2520^2;
%e family 6 is A082994(j) * 100^k: 8670000 * 768 = 81600^2;
%e family 7 is A323061(j) * 10^(2k+1): 5476580 * 856745 = 2166110^2.
%p revdigs:= proc(n) local L,i;
%p L:= convert(n,base,10);
%p add(L[-i]*10^(i-1),i=1..nops(L))
%p end proc:
%p filter:= n -> issqr(n*revdigs(n)):
%p select(filter, [$0..1000]);# _Robert Israel_, Feb 09 2019
%t Select[Range[0, 535], IntegerQ@ Sqrt[# IntegerReverse@ #] &] (* _Michael De Vlieger_, Feb 03 2019 *)
%o (PARI) isok(n) = issquare(n*fromdigits(Vecrev(digits(n)))); \\ _Michel Marcus_, Feb 04 2019
%Y Cf. A002113, A070760, A062917, A035090, A082994, A322835, A323061.
%Y Cf. A083406, A083407, A083408, A117281 (Squares = k * rev(k) in at least two ways).
%K nonn,base
%O 1,3
%A _Bernard Schott_, Feb 02 2019
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