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