A115442 -id:A115442 - cp's OEIS frontend

A115431 Numbers k such that the concatenation of k with k-2 gives a square.

6, 5346, 8083, 10578, 45531, 58626, 2392902, 2609443, 7272838, 51248898, 98009803, 159728062051, 360408196038, 523637103531, 770378933826, 998000998003, 1214959556998, 1434212848998, 3860012299771, 4243705560771

Offset: 1

Giovanni Resta, Jan 24 2006

From Robert Israel, Feb 20 2019: (Start) The same as A116117 and A116135 (see link).
So there are two equivalent definitions: numbers k such that k concatenated with k-6 gives the product of two numbers which differ by 4; and numbers k such that k concatenated with k-3 gives the product of two numbers which differ by 2.
For each k >= 1, 10^(4*k)-2*10^(3*k)+10^(2*k)-2*10^k+3 is a term.
If k is a term and k-2 has length m, then all prime factors of 10^m+1 must be congruent to 1 or 3 (mod 8).  In particular, we can't have m == 2 (mod 4) or m == 3 (mod 6), as in those cases 10^m+1 would be divisible by 101 or 7 respectively. (End)

			8083_8081 = 8991^2.
98009803_98009800 = 98999900 * 98999902, where _ denotes
concatenation

Robert Israel, Table of n, a(n) for n = 1..1312
Robert Israel, Proof that A115431, A116117 and A116135 are the same

Cf. A030465, A102567, A115426, A115437, A115428, A115429, A115430, A115432, A115433, A115434, A115435, A115436, A115442.

f:= proc(n) local S;
  S:= map(t -> rhs(op(t))^2 mod 10^n+2, [msolve(x^2+2,10^n+1)]);
  op(sort(select(t -> t-2 >= 10^(n-1) and t-2 < 10^n and issqr(t-2 + t*10^n), S)))
end proc:
seq(f(n),n=1..20); # Robert Israel, Feb 20 2019

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.

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

Original entry on oeis.org

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.

cp's OEIS Frontend

A115431 Numbers k such that the concatenation of k with k-2 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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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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