A106497 -id:A106497 - cp's OEIS frontend

A102567 Numbers k such that the concatenation of k with itself is a biperiod square.

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

Offset: 1

C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 15 2005

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
From Jianing Song, Nov 01 2024: (Start)
Numbers 10^(k-1) <= a <= 10^k - 1 such that a*(10^k + 1) is a square. Note that 10^k + 1 must be nonsquarefree, i.e., k is in A086982, otherwise a must be divisible by 10^k + 1, which is impossible.
Let v(p,m) be the p-adic valuation of m.
 - If p is not in A045616, then v(p,10^k+1) = r > 0 if and only if v(p,gcd(n,10^k+1)) = r-1.
 - If p is in A045616, let e be the multiplicative order of 10 modulo p, then v(p,10^k+1) > 0 if and only if e is even and k is an odd multiple of e/2, in which case v(p,10^k+1) = v(p,10^e-1) + v(p,k) = v(p,10^e-1) + v(p,gcd(k,10^k+1)).
This helps to find the terms. (End)

			13223140496 concatenated with 13223140496 is 1322314049613223140496 = 36363636364^2.
40495867769 is in the sequence because writing it twice gives the square number 4049586776940495867769 = 63636363637^2.

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.

David W. Wilson, Table of n, a(n) for n = 1..1098
Dr Barker, Can Numbers Like These Be Square?, YouTube video, 2023.
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.

Cf. A092118, A054214, A116163, A116136, A116279.

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:

A102567L[n_] := Catenate@Table[Module[{fac = FactorInteger[10^k + 1], min}, If[Max@fac[[All, -1]] == 1, {}, min = Times @@ Cases[fac, {a_, A102567L%5B30%5D%20(*%20_JungHwan%20Min">?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 *)

p = [3, 487, 56598313]; \\ A045616
b(n) = my(d = gcd(n, lift(Mod(10,n)^n)+1), s = 1); for(j=1, #p, my(e = znorder(Mod(10, p[j]))); if((e % 2 == 0) && (n % (e/2) == 0) && (n/(e/2) % 2 == 1), my(v = valuation(d, p[j])); d /= p[j]^v; s *= p[j]^((v+valuation(10^e-1, p[j]))\2))); my(f = factor(d)); for(i=1, #f~, s *= f[i,1]^((f[i,2]+1)\2)); s; \\ giving s such that 10^n + 1 = s^2*t where t is squarefree, considering only the three already-known terms of A045616
A102567_length_n(n) = my(t = (10^n+1)/b(n)^2, lowlim = 1+sqrtint(10^(n-1)\t), uplim = sqrtint((10^n-1)\t)); vector(uplim-lowlim+1, i, (lowlim-1+i)^2 * t) \\ terms of the form a^2*t such that 10^(n-1) <= a^2*t <= 10^n - 1
\\ Jianing Song, Nov 01 2024

from itertools import count, islice
from sympy import sqrt_mod
def A102567_gen(): # generator of terms
    for j in count(0):
        b = 10**j
        a = b*10+1
        for k in sorted(sqrt_mod(0,a,all_roots=True)):
            if a*b <= k**2 < a*(a-1):
                yield k**2//a
A102567_list = list(islice(A102567_gen(),10)) # Chai Wah Wu, Feb 19 2024

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

A115556 Numbers whose square is the concatenation of two numbers 9*m and m.

Original entry on oeis.org

12857142857142857142857142857142857143, 25714285714285714285714285714285714286, 117391304347826086956521739130434782608695652173913043478261

Offset: 1

Giovanni Resta, Jan 25 2006

a(4)=156521739130434782608695652173913043478260869565217391304348.
From Robert Israel, Aug 24 2023: (Start)
If 9 * 10^d + 1 = a^2 * b with a > 1, then a * b * c is a term if a^2/(90 + 10^(1-d)) < c^2 < a^2/(9 + 10^(-d)).  For example, 9 * 10^d + 1 is divisible by 7^2 for d == 37 (mod 42), and then (9 * 10^d + 1)/7 and 2*(9 * 10^d + 1)/7 are terms.  In particular, the sequence is infinite. (End)

Robert Israel, Table of n, a(n) for n = 1..12

Cf. A009474, A102567, A106497, A115527 - A115555.

F:= proc(d) local R,F,t,b,r,q,s,m0,x0,k;
     R:= NULL;
     F:= ifactors(9*10^d+1)[2];
     b:= mul(t[1]^floor(t[2]/2),t=F);
     for r in numtheory:-divisors(b) do
       x0:= (9*10^d+1)/r;
       m0:= x0/r;
       for k from ceil(sqrt(10^(d-1)/m0)) to floor(sqrt(10^d/m0)) do
         R:= R, x0*k;
       od
     od;
       R
end proc:
sort(map(F, [$1..90])); # Robert Israel, Aug 24 2023

Definition modified by Georg Fischer, Jul 26 2019

A115527 Numbers k such that the concatenation of k with 2*k gives a square.

Original entry on oeis.org

3, 8163265306122448979592, 18367346938775510204082, 32653061224489795918368, 504986744097967428355005681100871102133568993813912384800

Offset: 1

Giovanni Resta, Jan 25 2006

			3_6 = 6^2.

Cf. A102567, A106497, A115528 - A115556.

A115439 Numbers m such that the square of m is the concatenation of two numbers k and k+5.

Original entry on oeis.org

4, 7, 45, 56, 38163, 61838, 83618, 346980, 653021, 950051, 8647555, 9534265, 8167822283, 9007920992, 9209900792, 9950000501, 4737445289221, 4990568257187, 5009431742814, 5262554710780, 8373808925585, 8626931893551, 34323166122692, 34532758615690, 49625657225895, 49835249718893

Offset: 1

Giovanni Resta, Jan 24 2006

All numbers of the form f(n)=9(n).5.0(2n).5.0(n-1).1 where n>0 are in the sequence because if k(n)=9(n).0(n).25.0(n-1).9(n).6 then f(n)^2=k(n).(k(n)+5). For example f(2)=9950000501; k(2)=9900250996 and f(2)^2=9950000501^2=9900250996.9900251001 =k(2).(k(2)+5). - Farideh Firoozbakht, Nov 26 2006

m^2 = (k)|(k+5) = (k)|(k) + 5 = (10^q + 1)*k + 5 where | denotes concatenation and q is the number of digits of k gives a nonlinear equation that can be solved using the solver below. - David A. Corneth, Jan 02 2021

			38163^2 = 14564_14569.

David A. Corneth, Table of n, a(n) for n = 1..3733
Dario A. Alpern, Generic two integer variable equation solver

Cf. A030467, A057934, A106497, A115428, A115427, A115438, A115440, A115441, A115442, A115443, A115444, A115445, A115446, A115447.

More terms from David A. Corneth, Jan 02 2021

A115446 Numbers k such that the square of k is the concatenation of two numbers m and m-8.

Original entry on oeis.org

4623, 5378, 7981, 34953, 46866, 53135, 65048, 7056187, 9783460, 43176671, 56823330, 97999801, 447255476453, 552744523548, 755424659535, 799319866014, 997999998001, 4297663349524, 5702336650477, 6971253996228, 7574200549228, 8843117894979, 3505613322543666, 3757750389995601, 3948262973033353

Offset: 1

Giovanni Resta, Jan 25 2006

			18642249_18642241 = 43176671^2.

David A. Corneth, Table of n, a(n) for n = 1..1730

Cf. A030467, A106497, A115435, A115427, A115438, A115439, A115440, A115441, A115442, A115443, A115444, A115445, A115447.

More terms from David A. Corneth, Jan 02 2021

A115427 Numbers k such that k^2 is the concatenation of two numbers m and m+2.

Original entry on oeis.org

8874, 9011, 83352842, 99000101, 329767122288, 670232877713, 738226276373, 933006600341, 999000001001, 3779410975143115, 3872816717528067, 4250291784692550, 4278630943941867, 4372036686326819, 4749511753491302

Offset: 1

Giovanni Resta, Jan 24 2006

			9011^2 = 8119_8121.

Cf. A030467, A106497, A115426, A115438, A115439, A115440, A115441, A115442, A115443, A115444, A115445, A115446, A115447.

A115438 Numbers whose square is the concatenation of two numbers k and k+4.

Original entry on oeis.org

2, 310, 453, 548, 691, 856, 4382, 5619, 72730, 346533, 653468, 9090908, 94117646, 334665333, 336032387, 378253328, 390977442, 439928491, 483516486, 516483515, 560071510, 609022559, 621746673, 663967614, 665334668

Offset: 1

Giovanni Resta, Jan 24 2006

From Farideh Firoozbakht, Nov 26 2006: (Start)
1. All numbers of the form f(n)=3(n).4.6(n).5.3(n+1) are in the sequence because if k(n)=1(n).2.0(n+1).8(n).5 then f(n)^2= k(n).(k(n)+4). For example f(3)=333466653333; k(3)=111200008885 and f(3)^2=333466653333^2=k(3).(k(3)+4)=111200008885.111200008889.
2. All numbers of the form g(n)=6(n).5.3(n).4.6(n).8 are in the sequence because g(0)=548 is in the sequence(548^2=300.304) and for n>0 if h(n)=4(n).2.6(n-1).70.2(n).0 then g(n)^2=h(n).(h(n)+4). For example g(5)=666665333334666668; h(5)=444442666670222220 and g(5)^2=h(5).(h(5)+4)=444442666670222220.444442666670222224. (End)

			120085_120089 = 346533^2.

Cf. A030467, A106497, A115437, A115427, A115439, A115440, A115441, A115442, A115443, A115444, A115445, A115446, A115447.

The initial "2" (which is admittedly somewhat dubious) added by N. J. A. Sloane, Aug 13 2008

A115440 Numbers whose square is the concatenation of two numbers k and k+8.

Original entry on oeis.org

7747, 8021, 33294318, 66705683, 98000201, 340465755425, 476452552745, 523547447256, 659534244576, 866013200681, 998000002001, 3695104677080134, 3755782995538768, 4198081170077531, 4803478892324966, 5196521107675035

Offset: 1

Giovanni Resta, Jan 24 2006

			6001_6009 = 7747^2.

Cf. A030467, A106497, A115429, A115427, A115438, A115439, A115441, A115442, A115443, A115444, A115445, A115446, A115447.

A115441 Numbers whose square is the concatenation of two numbers k and k+9.

Original entry on oeis.org

465, 536, 718, 822, 3428, 6573, 90907, 980202, 3636361, 6363640, 41176468, 58823533, 413533838, 426573430, 428571426, 432620009, 567379992, 571428575, 573426571, 586466163, 686261111, 725274729, 727272725, 731321308

Offset: 1

Giovanni Resta, Jan 24 2006

			82640_82649 = 90907^2.

Cf. A030467, A106497, A115430, A115427, A115438, A115439, A115440, A115442, A115443, A115444, A115445, A115446, A115447.

A115442 Numbers whose square is the concatenation of two numbers k and k-2.

Original entry on oeis.org

8, 7312, 8991, 32524, 67477, 76568, 4891730, 5108271, 8528094, 71588336, 98999901, 399659933007, 600340066994, 723627738227, 877712329768, 998999999001, 3485626998114, 3787100274614, 6212899725387, 6514373001887

Offset: 1

Giovanni Resta, Jan 24 2006

			8083_8081 = 8991^2.

Cf. A030467, A106497, A115431, A115427, A115438, A115439, A115440, A115441, A115443, A115444, A115445, A115446, A115447.

Select[Table[Sqrt[k*10^IntegerLength[k]+k-2],{k,4,86*10^5}],IntegerQ] (* The program generates the first 9 terms of the sequence. *) (* Harvey P. Dale, Nov 02 2024 *)

cp's OEIS Frontend

A102567 Numbers k such that the concatenation of k with itself is a biperiod square.

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A115556 Numbers whose square is the concatenation of two numbers 9*m and m.

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A115527 Numbers k such that the concatenation of k with 2*k gives a square.

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A115439 Numbers m such that the square of m is the concatenation of two numbers k and k+5.

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A115446 Numbers k such that the square of k is the concatenation of two numbers m and m-8.

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A115427 Numbers k such that k^2 is the concatenation of two numbers m and m+2.

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A115438 Numbers whose square is the concatenation of two numbers k and k+4.

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A115440 Numbers whose square is the concatenation of two numbers k and k+8.

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A115441 Numbers whose square is the concatenation of two numbers k and k+9.

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A115442 Numbers whose square is the concatenation of two numbers k and k-2.

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