login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A306273 Numbers k such that k * rev(k) is a square, where rev=A004086, decimal reversal. 5
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 100, 101, 111, 121, 131, 141, 144, 151, 161, 169, 171, 181, 191, 200, 202, 212, 222, 232, 242, 252, 262, 272, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
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.
From Chai Wah Wu, Feb 18 2019: (Start)
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.
The same argument shows that numbers corresponding to axaxa, axaxaxa, ... are also terms.
For example, since 528 is a term, so are 528528, 5280528, 52800528, 5280052800528, etc.
(End)
REFERENCES
C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, NY. (1966), pp. 88-89.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, Revised edition (1997), p. 168.
LINKS
Bernard Schott, Sequences and Families
Eric Weisstein's World of Mathematics, Reversal
EXAMPLE
One example for each family:
family 1 is A002113: 323 * 323 = 323^2;
family 2 is A035090: 169 * 961 = 13^2 * 31^2 = 403^2;
family 3 is A082994: 288 * 882 = (2*144) * (2*441) = 504^2;
family 4 is A002113(j) * 100^k: 75700 * 757 = 7570^2;
family 5 is A035090(j) * 100^k: 44100 * 144 = 2520^2;
family 6 is A082994(j) * 100^k: 8670000 * 768 = 81600^2;
family 7 is A323061(j) * 10^(2k+1): 5476580 * 856745 = 2166110^2.
MAPLE
revdigs:= proc(n) local L, i;
L:= convert(n, base, 10);
add(L[-i]*10^(i-1), i=1..nops(L))
end proc:
filter:= n -> issqr(n*revdigs(n)):
select(filter, [$0..1000]); # Robert Israel, Feb 09 2019
MATHEMATICA
Select[Range[0, 535], IntegerQ@ Sqrt[# IntegerReverse@ #] &] (* Michael De Vlieger, Feb 03 2019 *)
PROG
(PARI) isok(n) = issquare(n*fromdigits(Vecrev(digits(n)))); \\ Michel Marcus, Feb 04 2019
CROSSREFS
Cf. A083406, A083407, A083408, A117281 (Squares = k * rev(k) in at least two ways).
Sequence in context: A071242 A044959 A353181 * A141033 A044821 A297148
KEYWORD
nonn,base
AUTHOR
Bernard Schott, Feb 02 2019
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 21 23:01 EDT 2024. Contains 371886 sequences. (Running on oeis4.)