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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A115436 Numbers k such that the concatenation of k with k-9 gives a square.

Original entry on oeis.org

50, 5234, 9410, 638370, 994010, 12477933, 41829698, 99940010, 1087279650, 4492494893, 6226356365, 7765453730, 9999400010, 806057802450, 842377434050, 960398039610, 999994000010, 21338126513658, 24752544267698
Offset: 1

Views

Author

Giovanni Resta, Jan 25 2006

Keywords

Examples

			638370_638361 = 798981^2.

Crossrefs

Cf. A030465, A102567, A115426, A115437, A115428, A115429, A115430, A115431, A115432, A115433, A115434, A115435, A115447.

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 *)

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

Views

Author

Giovanni Resta, Jan 24 2006

Keywords

Examples

			8083_8081 = 8991^2.

Crossrefs

Cf. A030467, A106497, A115431, A115427, A115438, A115439, A115440, A115441, A115443, A115444, A115445, A115446, A115447.

Programs

  • Mathematica
    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 *)

A116136 Numbers k such that k concatenated with k-3 gives the product of two numbers which differ by 4.

Original entry on oeis.org

9, 99, 183, 328, 528, 715, 999, 6099, 9999, 13224, 40495, 99999, 106755, 453288, 999999, 2066115, 2975208, 9999999, 22145328, 28027683, 99999999, 110213248, 110667555, 147928995, 178838403, 226123528, 275074575, 333052608, 378698224, 445332888, 446245635, 518348515
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Comments

Also numbers k such that k concatenated with itself gives the product of two numbers which differ by 2.

Examples

			8315420899//8315420896 = 9118892968 * 9118892972, where // denotes concatenation.

Links

Crossrefs

Cf. A116129, A115431, A116099, A115426, A116267.
Cf. also A102567, A116154, A116130, A116286.

Programs

  • Python
    from itertools import count, islice
from sympy import sqrt_mod
def A116136_gen(): # generator of terms
    for j in count(0):
        b = 10**j
        a = b*10+1
        for k in sorted(sqrt_mod(1,a,all_roots=True)):
            if a*(b+3) <= k**2-1 < a*(a+2):
                yield (k**2-1)//a
A116136_list = list(islice(A116136_gen(),40)) # Chai Wah Wu, Feb 19 2024

Extensions

Edited by N. J. A. Sloane, Apr 15 2007
a(29)-a(32) from Chai Wah Wu, Feb 19 2024

A116104 Numbers k such that k concatenated with k-8 gives the product of two numbers which differ by 4.

Original entry on oeis.org

65, 6653, 9605, 218413, 283720, 996005, 58446925, 99960005, 6086712229, 7385370133, 8478948853, 9999600005, 120178240093, 161171620229, 358247912200, 426843573160, 893417179213, 999996000005, 23376713203604
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Comments

Duplicate of A115432. See proof at A115432. - Robert Israel, Sep 13 2023

Examples

			9999600005//9999599997 = 9999799999 * 9999800003, where // denotes concatenation.

Crossrefs

Cf. A116096, A116103, A116105, A115431, A116235.

A116121 Numbers k such that k concatenated with k-5 gives the product of two numbers which differ by 2.

Original entry on oeis.org

65, 6653, 9605, 218413, 283720, 996005, 58446925, 99960005, 6086712229, 7385370133, 8478948853, 9999600005, 120178240093, 161171620229, 358247912200, 426843573160, 893417179213, 999996000005, 23376713203604
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Comments

Duplicate of A115432. See proof at A115432. - Robert Israel, Sep 13 2023

Examples

			9999600005//9999600000 = 9999800000 * 9999800002, where // denotes concatenation.

Crossrefs

Cf. A116116, A116120, A116122, A115431, A116252.

A116116 Numbers k such that k concatenated with k-6 gives the product of two numbers which differ by 2.

Original entry on oeis.org

21, 30, 902406, 959721, 6040059046, 6242406405, 9842410005, 9900249006, 15033519988494, 17250863148969, 22499666270469, 27632040031654, 34182546327286, 37487353123861, 52213551379230, 74230108225630
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Examples

			9900249006//9900249000 = 9949999500 * 9949999502, where // denotes concatenation.

Crossrefs

Cf. A116102, A116115, A115431, A116121, A116247.

A116118 Numbers k such that k concatenated with k-6 gives the product of two numbers which differ by 5.

Original entry on oeis.org

10, 12, 100, 102, 132, 406, 510, 852, 1000, 1002, 7930, 10000, 10002, 66942, 100000, 100002, 113322, 440056, 1000000, 1000002, 5289256, 10000000, 10000002, 58477510, 100000000, 100000002, 111333222, 164378892, 183673470
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Examples

			1000000000//999999994 = 999999998 * 1000000003, where // denotes concatenation.

Crossrefs

Cf. A116105, A115431, A116119, A116124, A116249.

A116248 n times n+4 gives the concatenation of two numbers m and m-6.

Original entry on oeis.org

7310, 8989, 32522, 67475, 76566, 4891728, 5108269, 8528092, 71588334, 98999899, 399659933005, 600340066992, 723627738225, 877712329766, 998999998999, 3485626998112, 3787100274612, 6212899725385, 6514373001885
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Examples

			98999899 * 98999903 = 98009803//98009797, where // denotes
concatenation.

Crossrefs

Cf. A115431, A116235, A116247, A116249, A116254.

A116266 n times n+2 gives the concatenation of two numbers m and m-3.

Original entry on oeis.org

7, 7311, 8990, 32523, 67476, 76567, 4891729, 5108270, 8528093, 71588335, 98999900, 399659933006, 600340066993, 723627738226, 877712329767, 998999999000, 3485626998113, 3787100274613, 6212899725386, 6514373001886
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Examples

			98999900 * 98999902 = 98009803//98009800, where // denotes
concatenation.

Crossrefs

Cf. A115431, A116252, A116265, A116267, A116273.
Previous Showing 11-20 of 21 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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