|
%I #36 Feb 19 2024 21:45:46
%S 13223140496,20661157025,29752066116,40495867769,52892561984,
%T 66942148761,82644628100,183673469387755102041,326530612244897959184,
%U 510204081632653061225,734693877551020408164
%N Numbers k such that the concatenation of k with itself is a biperiod square.
%C Also, numbers N associated with A106497.
%C 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
%D 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.
%D 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.
%H David W. Wilson, <a href="/A102567/b102567.txt">Table of n, a(n) for n = 1..1098</a>
%H Dr Barker, <a href="https://www.youtube.com/watch?v=c1peEN5Q-0c">Can Numbers Like These Be Square?</a>, YouTube video, 2023.
%H Andrew Bridy, Robert J. Lemke Oliver, Arlo Shallit, and Jeffrey Shallit, <a href="https://arxiv.org/abs/1707.03894">The Generalized Nagell-Ljunggren Problem: Powers with Repetitive Representations</a>, preprint arXiv:1707.03894 [math.NT], July 14 2017.
%e 13223140496 concatenated with 13223140496 is 1322314049613223140496 = 36363636364^2.
%e 40495867769 is in the sequence because writing it twice gives the square number 4049586776940495867769 = 63636363637^2.
%p 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:
%t 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 *)
%t A102567Q = IntegerQ@Sqrt@FromDigits[Join[#, #] &@IntegerDigits[#]] & (* _JungHwan Min_, Dec 11 2016 *)
%o (Python)
%o from itertools import count, islice
%o from sympy import sqrt_mod
%o def A102567_gen(): # generator of terms
%o for j in count(0):
%o b = 10**j
%o a = b*10+1
%o for k in sorted(sqrt_mod(0,a,all_roots=True)):
%o if a*b <= k**2 < a*(a-1):
%o yield k**2//a
%o A102567_list = list(islice(A102567_gen(),10)) # _Chai Wah Wu_, Feb 19 2024
%Y Cf. A092118, A054214, A116163, A116136, A116279.
%K easy,nonn,base
%O 1,1
%A C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 15 2005
%E Entry revised by _N. J. A. Sloane_, Nov 14 2006 and also Nov 27 2006
%E Definition edited and reference added by _William Rex Marshall_, Nov 12 2010
|
