A115430 -id:A115430 - cp's OEIS frontend

A115437 Numbers m such that the concatenation of m with m+4 gives a square.

96, 205, 300, 477, 732, 1920, 3157, 52896, 120085, 427020, 8264460, 88581312, 112000885, 112917765, 143075580, 152863360, 193537077, 233788192, 266755221, 313680096, 370908477, 386568925, 440852992, 442670220

Offset: 1

Giovanni Resta, Jan 24 2006

From Farideh Firoozbakht, Nov 26 2006: (Start)
a(n).(a(n)+4) = A115438^2 where "." denotes concatenation.
All numbers of the form f(j) = 4{j}.2.6{j-1}.70.2{j}.0 where each expression in braces denotes the multiplicity of the digit preceding the expression (e.g., "4{j}" means that the digit "4" appears j times consecutively) and where j > 0 are in the sequence because if k(j) = 6{j}.5.3{j}.4.6{j}.8 then k(j)^2 = f(j).(f(j)+4). For example, f(4) = 444426667022220, k(4) = 666653333466668, and k(4)^2 = 666653333466668^2 = f(4).(f(4)+4) = 444426667022220.444426667022224.
All numbers of the form f(j) = 1{j}.2.0{j+1}.8{j}.5 where j > -1 are in the sequence because if k(j) = 3{j}.4.6{j}.5.3{j+1} then k(j)^2 = f(j).(f(j)+4). For example, f(5) = 111112000000888885, k(5) = 333334666665333333, and k(5)^2 = 333334666665333333^2 = f(5).(f(5)+4) = 111112000000888885.111112000000888889. (End)

			Using "." to denote concatenation, 120085.120089 = 346533^2.

Cf. A030465, A102567, A115426, A115428, A115429, A115430, A115431, A115432, A115433, A115434, A115435, A115436, A115438.

Select[Range[10^5],IntegerQ@Sqrt@FromDigits@Flatten[IntegerDigits/@{#,#+4}]&] (* Giorgos Kalogeropoulos, Jul 27 2021 *)

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.

A116172 Numbers k such that k concatenated with k+2 gives the product of two numbers which differ by 5.

Original entry on oeis.org

2, 74, 59264, 510782, 906902, 81790664, 92776472, 10876856041862, 11796926254874, 18332259798794, 18388650720624, 32624670587648, 32699883214248, 43103618706398, 44916698243804, 66132258426302

Offset: 1

Giovanni Resta, Feb 06 2006

Also numbers k such that k concatenated with k+6 gives the product of two numbers which differ by 3.
Also numbers k such that k concatenated with k+8 gives the product of two numbers which differ by 1.
If k+2 and k-4 have the same number of digits, then k is also in A116132 because k//k+2 = 10^d*k + k + 2 = m*(m+5) then implies k//k-4 = 10^d*k + k - 4 = m*(m+5) - 6 = (m-1)*(m+6). - R. J. Mathar, Aug 10 2008

			92776472//92776474 = 96320542 * 96320547, where // denotes concatenation.
92776472//92776480 = 96320544 * 96320545.
92776472//92776478 = 96320543 * 96320546.

Cf. A116158, A116171, A116173, A116179, A116303.

Cf. A116185, A115430, A116205, A116343, A116200, A116330.

Edited by N. J. A. Sloane, Apr 15 2007

A116193 Numbers k such that k concatenated with k+5 gives the product of two numbers which differ by 5.

Original entry on oeis.org

11, 45, 18281, 32769, 56891, 180689, 330539, 959481, 1850201, 3247409, 4940219, 2425563239, 2575561739, 6003563495, 7245212645, 7770160145, 4983798265289, 5049762270381, 5534298528989, 5603798594169, 21894082450101

Offset: 1

Giovanni Resta, Feb 06 2006

Also numbers k such that k concatenated with k+9 gives the product of two numbers which differ by 3. For proof that this is the same sequence compare A116133.

Also numbers k such that k concatenated with k-9 gives the product of two numbers which differ by 9.

			7770160145//7770160136 = 8814851183 * 8814851192, where // denotes concatenation.

Cf. A116205, A116187, A116349.
Cf. A116179, A115430, A116194, A116200, A116324.
Cf. A116099, A116109, A116231.

Edited by N. J. A. Sloane, Apr 12 2007

A116168 Numbers k such that k concatenated with k+1 gives the product of two numbers which differ by 8.

Original entry on oeis.org

19, 32, 16284704, 35576083, 15764836187996024260119639732979, 19807200907254352332962649366152, 20298517078413563250826300137112, 30190765850423053042937262322867, 30796637697589506772859224996627

Offset: 1

Giovanni Resta, Feb 06 2006

Also numbers k such that k concatenated with k+8 gives the product of two numbers which differ by 6.

			35576083//35576084 = 59645686 * 59645694, where // denotes concatenation.
35576083//35576091 = 59645687 * 59645693.

Cf. A116161, A116167, A116169, A116174, A116299.

Cf. A116208, A115430, A116215, A116345.

Edited by N. J. A. Sloane, Apr 15 2007

A116173 Numbers k such that k concatenated with k+2 gives the product of two numbers which differ by 6.

Original entry on oeis.org

6752089, 6448802889351008245, 18894512461523256139943105859903480218905, 31958875438439894736354375209245786214798

Offset: 1

Giovanni Resta, Feb 06 2006

Also numbers k such that k concatenated with k+7 gives the product of two numbers which differ by 4.

If k+2 and k-5 have the same number of digits, the k is also in A116126, because k//k+2 = 10^d*k + k + 2 = m*(m+6) then implies a representation k//k-5 = 10^d*k + k - 5 = m*(m+6)-7 = (m-1)*(m+7). - R. J. Mathar, Aug 10 2008

			6752089//6752091 = 8217107 * 8217113, where // denotes concatenation.
6752089//6752096 = 8217108 * 8217112.

Cf. A116159, A116172, A116174, A116187, A116304.

Cf. A115430, A116205, A116207, A116187, A116337.

Edited by N. J. A. Sloane, Apr 12 2007

A116323 n times n+4 gives the concatenation of two numbers m and m+5.

Original entry on oeis.org

463, 534, 716, 820, 3426, 6571, 90905, 980200, 3636359, 6363638, 41176466, 58823531, 413533836, 426573428, 428571424, 432620007, 567379990, 571428573, 573426569, 586466161, 686261109, 725274727, 727272723, 731321306

Offset: 1

Giovanni Resta, Feb 06 2006

Cf. A115430, A116317, A116322, A116324, A116337.

A116344 Numbers k such that k*(k+2) gives the concatenation of two numbers m and m+8.

Original entry on oeis.org

464, 535, 717, 821, 3427, 6572, 90906, 980201, 3636360, 6363639, 41176467, 58823532, 413533837, 426573429, 428571425, 432620008, 567379991, 571428574, 573426570, 586466162, 686261110, 725274728, 727272724, 731321307

Offset: 1

Giovanni Resta, Feb 06 2006

Cf. A115430, A116335, A116343, A116345.

cp's OEIS Frontend

A115437 Numbers m such that the concatenation of m with m+4 gives a square.

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

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A116172 Numbers k such that k concatenated with k+2 gives the product of two numbers which differ by 5.

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A116193 Numbers k such that k concatenated with k+5 gives the product of two numbers which differ by 5.

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A116168 Numbers k such that k concatenated with k+1 gives the product of two numbers which differ by 8.

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A116173 Numbers k such that k concatenated with k+2 gives the product of two numbers which differ by 6.

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A116323 n times n+4 gives the concatenation of two numbers m and m+5.

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A116344 Numbers k such that k*(k+2) gives the concatenation of two numbers m and m+8.

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