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A306273 Numbers k such that k * rev(k) is a square, where rev=A004086, decimal reversal. 4
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

Robert Israel, Table of n, a(n) for n = 1..10000

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. A002113, A070760, A062917, A035090, A082994, A322835, A323061.

Cf. A083406, A083407, A083408, A117281 (Squares = k * rev(k) in at least two ways).

Sequence in context: A071242 A044959 A353181 * A141033 A044821 A297148

Adjacent sequences: A306270 A306271 A306272 * A306274 A306275 A306276

KEYWORD

nonn,base

AUTHOR

Bernard Schott, Feb 02 2019

STATUS

approved

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Last modified December 1 17:49 EST 2022. Contains 358475 sequences. (Running on oeis4.)