login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A082994 Numbers n such that all the following properties hold: (i) n*reverse(n) is a square; (ii) n != reverse(n); (iii) n and reverse(n) are not both squares; and (iv) n and reverse(n) have the same number of digits. 4
288, 528, 768, 825, 867, 882, 1584, 2178, 4851, 8712, 10989, 13104, 14544, 15984, 20808, 21978, 26208, 27648, 27848, 36828, 40131, 44541, 48139, 48951, 49686, 57399, 68694, 80262, 80802, 82863, 84672, 84872, 87912, 93184, 98901, 99375 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

These numbers are counterexamples to the following conjecture given in the Ogilvy-Anderson reference: "When an integer and its reversal are unequal, their product is never a square except when both are squares." This sequence excludes terms like 2200, i.e. 2200*22 = 48400.

Contains x*(10^k+1) for k >= 3 with x in {144, 169, 288, 441, 528, 768, 825, 867, 882, 961}. - Robert Israel, Jun 11 2018

A035090 U {this sequence} = A062917, with empty intersection. - Bernard Schott, Jan 04 2019

REFERENCES

C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, NY. (1966), pp. 88-89.

J. Earls, Mathematical Bliss, Pleroma Publications, 2009, pages 82-83. ASIN: B002ACVZ6O [From Jason Earls, Nov 22 2009]

LINKS

Robert Israel, Table of n, a(n) for n = 1..168

M. A. Rashid, M. A. Uppal, D. C. B. Marsh and A. Wayne, Product of a Number and Its Reverse, American Mathematical Monthly, vol. 64 (1957), p. 434, E-1243. - Felix Fröhlich, Jul 11 2014

EXAMPLE

a(5) = 867 because 867 * 768 = 665856 = 816^2.

MAPLE

revdigs:= proc(n) local L;

  L:= convert(n, base, 10);

  add(L[-i]*10^(i-1), i=1..nops(L))

end proc:

filter:= proc(n) local r;

  if issqr(n) then return false fi;

  r:= revdigs(n);

  r <> n and issqr(r*n) and not issqr(r);

end proc:

select(filter, [seq(seq(10*i+j, j=1..9), i=1..10^4)]); # Robert Israel, Jun 11 2018

MATHEMATICA

Select[Range[10^5], And[UnsameQ @@ {#1, #2}, IntegerQ@ Sqrt[#1 #2], AllTrue[{#1, #2}, ! IntegerQ@ Sqrt@ # &], SameQ @@ (IntegerLength@ {#1, #2})] & @@ {#, IntegerReverse@ #} &] (* Michael De Vlieger, Jan 04 2019 *)

CROSSREFS

Cf. A002113, A004086, A035090, A062917, A070760, A322835.

Sequence in context: A280515 A061831 A250774 * A127350 A158253 A292054

Adjacent sequences:  A082991 A082992 A082993 * A082995 A082996 A082997

KEYWORD

base,nonn

AUTHOR

Jason Earls, May 29 2003

EXTENSIONS

Name clarified by Bernard Schott, Jan 04 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 | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 16 14:47 EST 2019. Contains 320163 sequences. (Running on oeis4.)