A106497 -id:A106497 - cp's OEIS frontend

A115443 Numbers whose square is the concatenation of two numbers k and k-4.

81, 8157, 9801, 467347, 532654, 998001, 76450589, 99980001, 7801738415, 8593817623, 9208120793, 9999800001, 346667333467, 401461854015, 598538145986, 653332666534, 945207479453, 999998000001, 48349470735060

Offset: 1

Giovanni Resta, Jan 25 2006

			9605_9601 = 9801^2.

Cf. A030467, A106497, A115432, A115427, A115438, A115439, A115440, A115441, A115442, A115444, A115445, A115446, A115447.

A115444 Numbers whose square is the concatenation of two numbers k and k-5.

Original entry on oeis.org

46, 55, 949951, 979654, 7771781679, 7900890080, 9920892100, 9949999501, 38773083432317, 41534158410842, 47433813119408, 52566186880593, 58465841589159, 61226916567684, 72258945037435, 86156896546725

Offset: 1

Giovanni Resta, Jan 25 2006

			902406_902401 = 949951^2.

Cf. A030467, A106497, A115433, A115427, A115438, A115439, A115440, A115441, A115442, A115443, A115445, A115446, A115447.

A115445 Numbers whose square is the concatenation of two numbers k and k-7.

Original entry on oeis.org

9, 13, 3656545, 4565636, 5434365, 6343456, 3646962589704198389, 6353037410295801612, 9101508044249652935, 7903999111431765764698711045778, 9722180929613583946516892863960

Offset: 1

Giovanni Resta, Jan 25 2006

			4023943_4023936 = 6343456^2.

Cf. A030467, A106497, A115434, A115427, A115438, A115439, A115440, A115441, A115442, A115443, A115444, A115446, A115447.

A115447 Numbers whose square is the concatenation of two numbers k and k-9.

Original entry on oeis.org

71, 7235, 9701, 798981, 997001, 35324118, 64675883, 99970001, 3297392379, 6702607622, 7890726434, 8812181189, 9999700001, 897807218979, 917811219179, 979998999801, 999997000001, 46193210013657, 49751928874867

Offset: 1

Giovanni Resta, Jan 25 2006

			638370_638361 = 798981^2.

Cf. A030467, A106497, A115436, A115427, A115438, A115439, A115440, A115441, A115442, A115443, A115444, A115445, A115446.

A092118 Biperiod squares: square numbers whose digits repeat twice in order.

Original entry on oeis.org

1322314049613223140496, 2066115702520661157025, 2975206611629752066116, 4049586776940495867769, 5289256198452892561984, 6694214876166942148761, 8264462810082644628100, 183673469387755102041183673469387755102041

Offset: 1

Michael Mark, Dec 15 2004

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.

Author?, MMB message board "big square"
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. A102567, A106497.

f:=proc(n) local i,j,k; i:=cat(n,n); j:=convert(i,decimal,10); issqr(j); end;
with(numtheory): Digits:=50:for d from 1 to 22 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, ",tendp1*n*i^2) od: od: # C. Ronaldo

from itertools import count, islice
from sympy import sqrt_mod
def A092118_gen(): # generator of terms
    for j in count(0):
        b = 10**j
        a = b*10+1
        ab, aa = a*b, a*(a-1)
        for k in sorted(sqrt_mod(0,a,all_roots=True)):
            if ab <= (m:=k**2) < aa:
                yield m
A092118_list = list(islice(A092118_gen(),10)) # Chai Wah Wu, Mar 06 2024

Corrected and extended by C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 15 2005
Definition corrected and improved, reference and cross-reference added by William Rex Marshall, Nov 12 2010
Keyword base added by William Rex Marshall, Nov 12 2010

A115528 Numbers k such that k^2 is the concatenation of two numbers m and 2*m.

Original entry on oeis.org

6, 28571428571428571428572, 42857142857142857142858, 57142857142857142857144, 2247191011235955056179775280898876404494382022471910112360

Offset: 1

Giovanni Resta, Jan 25 2006

			6^2 = 3_6.

Cf. A102567, A106497, A115527 - A115556.

A115549 Numbers k such that the concatenation of k with 8*k gives a square.

Original entry on oeis.org

3, 12, 28, 63, 112, 278, 1112, 2778, 11112, 27778, 111112, 277778, 1111112, 2777778, 4938272, 7716050, 11111112, 12802888, 13151250, 13504288, 13862002, 14224392, 14591458, 14963200, 15339618, 15720712, 16106482, 16496928, 16892050, 17291848, 17696322, 18105472

Offset: 1

Giovanni Resta, Jan 25 2006

If k = 10*R_m + 2, with m >= 1, then the concatenation of k with 8*k equals (30*R_m + 6)^2, so A047855 \ {1,2} is a subsequence. - Bernard Schott, Apr 09 2022
Numbers k such that A009470(k) is a square. - Michel Marcus, Apr 09 2022
The numbers 28, 278, 2778, ..., 2*10^k + 7*(10^k - 1)/9 + 1, ..., k >= 1, are terms, because the concatenation forms the squares 28224 = 168^2, 2782224 = 1668^2, 277822224 = 16668^2, ..., (10^m + 2*(10^m - 1)/3 + 2)^2, m >= 2, ... - Marius A. Burtea, Apr 10 2022

			3_24 = 18^2.
11112_88896 = 33336^2.

Marius A. Burtea, Table of n, a(n) for n = 1..176

Cf. A002275, A047855, A009470, A102567, A106497, A115527-A115556.

[n:n in [1..20000000]|IsSquare(Seqint(Intseq(8*n) cat Intseq(n)))]; // Marius A. Burtea, Apr 10 2022

isok(k) = issquare(eval(Str(k, 8*k))); \\ Michel Marcus, Apr 09 2022

More terms from Marius A. Burtea, Apr 13 2022

A115529 Numbers k such that the concatenation of 2*k with k gives a square.

Original entry on oeis.org

1632653061224489796, 3673469387755102041, 6530612244897959184, 108166576527852893455922120064900, 130881557598702001081665765278529, 155759870200108166576527852893456, 182801514332071389940508382909681

Offset: 1

Giovanni Resta, Jan 25 2006

Numbers of the form k = a*b^2 where 10^(d-1) <= k < 10^d and (2*10^d+1)/a is a square. - Robert Israel, Jan 13 2021

			3265306122448979592_1632653061224489796 = 5714285714285714286^2.

Robert Israel, Table of n, a(n) for n = 1..36

Cf. A102567, A106497, A115527 - A115556.

f:= proc(d) local R,q,F,G,s,t,a,u,i;
   q:= 2*10^d+1;
   F:= ifactors(q)[2];
   G:= map(t -> [t[1],floor(t[2]/2)], F);
   s:= mul(t[1]^t[2],t=G);
   R:= NULL:
   for a in numtheory:-divisors(s) do
     u:= q/a^2;
     R:= R, seq(i^2*u, i=ceil(sqrt(10^(d-1)/u))..floor(sqrt((10^d-1)/u)))
   od;
   R
end proc:
seq(f(d),d=1..33); # Robert Israel, Jan 13 2021

A115531 Numbers k such that the concatenation of k with 3*k gives a square.

Original entry on oeis.org

816326530612244897959183673469388, 1836734693877551020408163265306123, 3265306122448979591836734693877552, 3746097814776274713839750260145681581685744016649323621228

Offset: 1

Giovanni Resta, Jan 25 2006

If 3+10^m is not squarefree, say 3+10^m = u^2*v where v is squarefree, then the terms with length m are t^2*v where 10^m > 3*t^2*v >= 10^(m-1). The first m for which 3+10^m is not squarefree are 34, 59, 60, 61, 67. - Robert Israel, Aug 07 2019

Since 3+10^m is divisible by 7^2 for m = 34 + 42*k, the sequence contains 4*(3+10^m)/49, 9*(3+10^m)/49 and 16*(3+10^m)/49 for such m, and in particular is infinite. - Robert Israel, Aug 08 2019

Robert Israel, Table of n, a(n) for n = 1..113

Cf. A102567, A106497, A115527 - A115556.

Res:= NULL:
for m from 1 to 67 do
if not numtheory:-issqrfree(3+10^m) then
   F:= select(t -> t[2]=1, ifactors(3+10^m)[2]);
   v:= mul(t[1], t=F);
   Res:= Res, seq(t^2*v, t = ceil(sqrt(10^(m-1)/(3*v))) .. floor(sqrt(10^m/(3*v))))
fi
od:
Res;  # Robert Israel, Aug 07 2019

A115555 Numbers k such that the concatenation of 9*k with k gives a square.

Original entry on oeis.org

1836734693877551020408163265306122449, 7346938775510204081632653061224489796, 15311909262759924385633270321361058601134215500945179584121, 27221172022684310018903591682419659735349716446124763705104

Offset: 1

Giovanni Resta, Jan 25 2006

Cf. A102567, A106497, A115527 - A115556.

cp's OEIS Frontend

A115443 Numbers whose square is the concatenation of two numbers k and k-4.

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A115444 Numbers whose square is the concatenation of two numbers k and k-5.

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A115445 Numbers whose square is the concatenation of two numbers k and k-7.

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A115447 Numbers whose square is the concatenation of two numbers k and k-9.

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A092118 Biperiod squares: square numbers whose digits repeat twice in order.

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A115528 Numbers k such that k^2 is the concatenation of two numbers m and 2*m.

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A115549 Numbers k such that the concatenation of k with 8*k gives a square.

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A115529 Numbers k such that the concatenation of 2*k with k gives a square.

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Maple

A115531 Numbers k such that the concatenation of k with 3*k gives a square.

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Maple

A115555 Numbers k such that the concatenation of 9*k with k gives a square.

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