

A102567


Numbers n such that n concatenated with itself is a biperiod square.


49



13223140496, 20661157025, 29752066116, 40495867769, 52892561984, 66942148761, 82644628100, 183673469387755102041, 326530612244897959184, 510204081632653061225, 734693877551020408164
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OFFSET

1,1


COMMENTS

Also, numbers N associated with A106497.
Also, numbers n such that n concatenated with n1 gives the product of two numbers which differ by 2. E.g. 13223140496//13223140495 = 36363636363 * 36363636365, where // denotes concatenation.  Giovanni Resta and Franklin T. AdamsWatters, Nov 13 2006


REFERENCES

R. Ondrejka, Problem 1130: Biperiod Squares, Journal of Recreational Mathematics, Vol. 14:4 (198182), 299. Solution by F. H. Kierstead, Jr., JRM, Vol. 15:4 (198283), 311312.


LINKS

David W. Wilson, Table of n, a(n) for n = 1..1098


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^(d1))/n)) to floor(sqrt((10^d1)/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


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



