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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)[2]: n:=mul(piecewise(tendp1fact[i][2] mod 2=1, tendp1fact[i][1], 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[30] (* 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.)