cp's OEIS Frontend

A306273 Numbers k such that k * rev(k) is a square, where rev=A004086, decimal reversal.

%I A306273 #56 Feb 16 2025 08:33:55
%S A306273 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 A306273 141,144,151,161,169,171,181,191,200,202,212,222,232,242,252,262,272,
%U A306273 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 A306273 Numbers k such that k * rev(k) is a square, where rev=A004086, decimal reversal.
%C A306273 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 A306273 From _Chai Wah Wu_, Feb 18 2019: (Start)
%C A306273 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 A306273 The same argument shows that numbers corresponding to axaxa, axaxaxa, ... are also terms.
%C A306273 For example, since 528 is a term, so are 528528, 5280528, 52800528, 5280052800528, etc.
%C A306273 (End)
%D A306273 C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, NY. (1966), pp. 88-89.
%D A306273 David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, Revised edition (1997), p. 168.
%H A306273 Robert Israel, <a href="/A306273/b306273.txt">Table of n, a(n) for n = 1..10000</a>
%H A306273 Bernard Schott, <a href="/A306273/a306273_2.pdf">Sequences and Families</a>
%H A306273 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Reversal.html">Reversal</a>
%e A306273 One example for each family:
%e A306273 family 1 is A002113: 323 * 323 = 323^2;
%e A306273 family 2 is A035090: 169 * 961 = 13^2 * 31^2 = 403^2;
%e A306273 family 3 is A082994: 288 * 882 = (2*144) * (2*441) = 504^2;
%e A306273 family 4 is A002113(j) * 100^k: 75700 * 757 = 7570^2;
%e A306273 family 5 is A035090(j) * 100^k: 44100 * 144 = 2520^2;
%e A306273 family 6 is A082994(j) * 100^k: 8670000 * 768 = 81600^2;
%e A306273 family 7 is A323061(j) * 10^(2k+1): 5476580 * 856745 = 2166110^2.
%p A306273 revdigs:= proc(n) local L,i;
%p A306273   L:= convert(n,base,10);
%p A306273   add(L[-i]*10^(i-1),i=1..nops(L))
%p A306273 end proc:
%p A306273 filter:= n -> issqr(n*revdigs(n)):
%p A306273 select(filter, [$0..1000]);# _Robert Israel_, Feb 09 2019
%t A306273 Select[Range[0, 535], IntegerQ@ Sqrt[# IntegerReverse@ #] &] (* _Michael De Vlieger_, Feb 03 2019 *)
%o A306273 (PARI) isok(n) = issquare(n*fromdigits(Vecrev(digits(n)))); \\ _Michel Marcus_, Feb 04 2019
%Y A306273 Cf. A002113, A070760, A062917, A035090, A082994, A322835, A323061.
%Y A306273 Cf. A083406, A083407, A083408, A117281 (Squares = k * rev(k) in at least two ways).
%K A306273 nonn,base
%O A306273 1,3
%A A306273 _Bernard Schott_, Feb 02 2019