A115443 -id:A115443 - cp's OEIS frontend

A115432 Numbers k such that the concatenation of k with k-4 gives a square.

65, 6653, 9605, 218413, 283720, 996005, 58446925, 99960005, 6086712229, 7385370133, 8478948853, 9999600005, 120178240093, 161171620229, 358247912200, 426843573160, 893417179213, 999996000005, 23376713203604

Offset: 1

Giovanni Resta, Jan 25 2006

The terms of this sequence (k//k-4 = m*m), A116104 (k//k-8 = m*(m+4)) and A116121 (k//k-5 = m*(m+2)) agree as long as the two concatenated numbers k and k-x have the same length. This condition is satisfied for the given terms of all three sequences. - Georg Fischer, Sep 12 2022
From Robert Israel, Sep 13 2023: (Start)
Numbers k of the form (y^2+4)/(10^d + 1) where 10^(d-1) <= k - 4 < 10^d and y is a square root of -4 mod (10^d + 1).
Includes 10^(2*d) - 4*10^d + 5 for all d >= 1, as the concatenation of this with 10^(2*d) - 4*10^d + 1 is 10^(4*d) - 4 * 10^(3*d) + 6 * 10^(2*d) - 4 * 10^d + 1 = (10^d - 1)^4.
This is the same sequence as A116104 and A116121.  The only possible differences would be if 10^(d-1) + 4 <= k <= 10^(d-1) + 7 or 10^d + 4 <= k <= 10^d + 7, so that k - 4 and k - 8 have different numbers of digits.
But in none of those cases can (10^d + 1)*k - 4 be a square:
If k = 10^(d-1) + 4 or 10^d + 4, (10^d + 1)*k - 4 == 6 (mod 9).
If k = 10^(d-1) + 5 or 10^d + 5, (10^d + 1)*k - 4 == 2 (mod 3).
If k = 10^(d-1) + 6 or 10^d + 6, (10^d + 1)*k - 4 == 2 (mod 10).
If k = 10^(d-1) + 7 or 10^d + 7, (10^d + 1)*k - 4 == 3 (mod 10). (End)

			9605_9601 = 9801^2.

Robert Israel, Table of n, a(n) for n = 1..2805

Cf. A030465, A102567, A115426, A115437, A115428, A115429, A115430, A115431, A115433, A115434, A115435, A115436, A115443.

Almost the same as A116104 and A116121.

f:= proc(d) uses NumberTheory; local m,r;
  m:= 10^d + 1;
  if QuadraticResidue(-4,m) = -1 then return NULL fi;
  r:= ModularSquareRoot(-4, m);
  op(sort(select(t -> t >= 10^(d-1)+4 and t < 10^d+4, map(t -> ((r*t mod m)^2+4)/m, convert(RootsOfUnity(2,m),list)))))
end proc:
map(f, [$1..20]); # Robert Israel, Sep 12 2023

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 *)

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.

cp's OEIS Frontend

A115432 Numbers k such that the concatenation of k with k-4 gives a square.

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Maple

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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Mathematica

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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