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 A102567 Numbers k such that the concatenation of k with itself is a biperiod square. 49
 13223140496, 20661157025, 29752066116, 40495867769, 52892561984, 66942148761, 82644628100, 183673469387755102041, 326530612244897959184, 510204081632653061225, 734693877551020408164 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Also, numbers N associated with A106497. Also, numbers k such that k concatenated with k-1 gives the product of two numbers which differ by 2. E.g., 13223140496//13223140495 = 36363636363 * 36363636365, where // denotes concatenation. - Giovanni Resta and Franklin T. Adams-Watters, Nov 13 2006 REFERENCES Andrew Bridy, Robert J. Lemke Oliver, Arlo Shallit, and Jeffrey Shallit, The Generalized Nagell-Ljunggren Problem: Powers with Repetitive Representations, Experimental Math, 28 (2019), 428-439. R. Ondrejka, Problem 1130: Biperiod Squares, Journal of Recreational Mathematics, Vol. 14:4 (1981-82), 299. Solution by F. H. Kierstead, Jr., JRM, Vol. 15:4 (1982-83), 311-312. LINKS David W. Wilson, Table of n, a(n) for n = 1..1098 Andrew Bridy, Robert J. Lemke Oliver, Arlo Shallit, and Jeffrey Shallit, The Generalized Nagell-Ljunggren Problem: Powers with Repetitive Representations, preprint arXiv:1707.03894 [math.NT], July 14 2017. EXAMPLE 13223140496 concatenated with 13223140496 is 1322314049613223140496 = 36363636364^2. 40495867769 is in the sequence because writing it twice gives the square number 4049586776940495867769 = 63636363637^2. MAPLE with(numtheory): Digits:=50:for d from 1 to 35 do tendp1:=10^d+1: tendp1fact:=ifactors(tendp1): n:=mul(piecewise(tendp1fact[i] mod 2=1, tendp1fact[i], 1), i=1..nops(tendp1fact)):for i from ceil(sqrt((10^(d-1))/n)) to floor(sqrt((10^d-1)/n)) do printf("%d, ", n*i^2) od: od: MATHEMATICA A102567L[n_] := Catenate@Table[Module[{fac = FactorInteger[10^k + 1], min}, If[Max@fac[[All, -1]] == 1, {}, min = Times @@ Cases[fac, {a_, _?OddQ} :> a]; Table[min s^2, {s, Ceiling@Sqrt[10^(k - 1)/min], Floor@Sqrt[(10^k - 1)/min]}]]], {k, n}]; A102567L (* JungHwan Min, Dec 11 2016 *) A102567Q = IntegerQ@Sqrt@FromDigits[Join[#, #] &@IntegerDigits[#]] & (* JungHwan Min, Dec 11 2016 *) CROSSREFS Cf. A092118, A054214, A116163, A116136, A116279. Sequence in context: A204096 A113639 A059122 * A016932 A016992 A017160 Adjacent sequences:  A102564 A102565 A102566 * A102568 A102569 A102570 KEYWORD easy,nonn,base,changed AUTHOR C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 15 2005 EXTENSIONS Entry revised by N. J. A. Sloane, Nov 14 2006 and also Nov 27 2006 Definition edited and reference added by William Rex Marshall, Nov 12 2010 STATUS approved

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Last modified June 22 17:18 EDT 2021. Contains 345388 sequences. (Running on oeis4.)