cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

Views

Author

Giovanni Resta, Jan 24 2006

Keywords

Comments

From Farideh Firoozbakht, Nov 26 2006: (Start)
1. a(n).(a(n)+4) = A115438^2 where "." denotes concatenation.
2. 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.
3. 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)

Examples

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

Crossrefs

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

Programs

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

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

Views

Author

Giovanni Resta, Jan 24 2006

Keywords

Examples

			6001_6009 = 7747^2.

Crossrefs

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

A116205 Numbers k such that k concatenated with k+7 gives the product of two numbers which differ by 3.

Original entry on oeis.org

1, 81, 1353, 3997, 7723, 23761, 26271, 76771, 1415683, 3890571, 8495497, 1066870443, 1239366513, 4198438981, 4534273891, 6502317141, 6918679731, 2199164200036329043, 2820114781174460091, 5500888421709400741
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Comments

Also numbers k such that k concatenated with k+9 gives the product of two numbers which differ by 1.
Also numbers k such that k concatenated with k-3 gives the product of two numbers which differ by 7.

Crossrefs

Cf. A116179, A116172, A115429, A116173, A116193, A116336.
Cf. A116132, A116099, A116139, A116152, A116269, A116348.

Extensions

Edited by N. J. A. Sloane, Apr 13 2007

A116107 Numbers k such that k concatenated with k-8 gives the product of two numbers which differ by 7.

Original entry on oeis.org

52, 63716866, 48793687600063875363014809897052, 60020753655608135708762056127156, 60446518621981165303188950156776, 71135436903815748345367595855336, 72876856643103028189103298533248
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Comments

Also numbers n such that n concatenated with n-2 gives the product of two numbers which differ by 5.

Examples

			63716866//63716864 = 79822843 * 79822848, where // denotes concatenation.

Crossrefs

Cf. A116099, A116106, A115429, A116112, A116238, A115426, A116146, A116276.

Extensions

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

A116187 Numbers k such that k concatenated with k+4 gives the product of two numbers which differ by 6.

Original entry on oeis.org

12, 43, 20440836, 30017751, 61336887, 52400871197755334426147587, 53651708763838760619655612, 56652002793835820319625612, 57952296063256269823192087, 17684775866714240650923831481623
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Comments

Also numbers k such that k concatenated with k+9 gives the product of two numbers which differ by 4. For proof that this is the same sequence compare A116133.

Examples

			61336887//61336891 = 78317867 * 78317873, where // denotes concatenation. 61336887//61336896 = 78317868 * 78317872.

Crossrefs

Cf. A116173, A115429, A116133, A116194, A116318.
Cf. A116193, A116180, A116350.

Extensions

Edited by N. J. A. Sloane, Apr 11 2007

A116185 Numbers k such that k concatenated with k+4 gives the product of two numbers which differ by 3.

Original entry on oeis.org

150, 186, 324, 376, 666, 2046, 3000, 82650, 100384, 466716, 1322316, 4049584, 67820074, 110003884, 135734074, 156502836, 196043286, 213017754, 238849000, 261405396, 289940826, 310507774, 365294050, 398891964, 446667216
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Comments

Also numbers k such that k concatenated with k+6 gives the product of two numbers which differ by 1.

Crossrefs

Cf. A116112, A115428, A115429, A116316, A116199, A116172, A116329.

Programs

  • Maple
    cc:=proc(x,y) local s: s:=proc(m) nops(convert(m,base,10)) end: x*10^s(y)+y: end: a:=proc(n) if type(sqrt(9+4*cc(n,n+4)),integer) then n else fi end: seq(a(n),n=1..500000); # very slow; cc yields the concatenation of x and y; - Emeric Deutsch, May 05 2007

Extensions

Edited by N. J. A. Sloane, Apr 14 2007

A116113 Numbers k such that k concatenated with k-7 gives the product of two numbers which differ by 8.

Original entry on oeis.org

95, 216, 287, 515, 675, 995, 1175, 4320, 9995, 82640, 99995, 960795, 999995, 1322312, 4049591, 9999995, 16955015, 34602080, 99999995, 171010235, 181964891, 183673467, 187160072, 321920055, 326530616, 328818032
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Examples

			8533818720//8533818713 = 9237867023 * 9237867031, where // denotes concatenation.

Crossrefs

Cf. A115429, A116112, A116114, A116126, A116244.

A116131 Numbers k such that k concatenated with k-4 gives the product of two numbers which differ by 6.

Original entry on oeis.org

20, 31, 14564, 38239, 69919, 120395, 426436, 902596, 7478020, 9090220, 6671332084, 8114264059, 8482227259, 9900250996, 2244338786836, 2490577152964, 2509440638591, 2769448208395, 7012067592220
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Examples

			9900250996//9900250992 = 9950000498 * 9950000504, where // denotes concatenation.

Crossrefs

Cf. A116125, A116130, A116132, A115429, A116262.

A116152 Numbers k such that k concatenated with k-1 gives the product of two numbers which differ by 7.

Original entry on oeis.org

9, 11, 45, 18281, 32769, 56891, 180689, 330539, 959481, 1850201, 3247409, 4940219, 2425563239, 2575561739, 6003563495, 7245212645, 7770160145, 4983798265289, 5049762270381, 5534298528989, 5603798594169
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Crossrefs

Cf. A116205, A115429, A116153, A116160, A116283.

A116159 Numbers k such that k concatenated with itself gives the product of two numbers which differ by 6.

Original entry on oeis.org

5, 95, 216, 287, 515, 675, 995, 1175, 4320, 9995, 82640, 99995, 960795, 999995, 1322312, 4049591, 9999995, 16955015, 34602080, 99999995, 171010235, 181964891, 183673467, 187160072, 321920055, 326530616, 328818032
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Crossrefs

Cf. A115429, A116158, A116160, A116173, A116290.
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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