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.

Showing 1-10 of 11 results. Next

A115426 Numbers k such that the concatenation of k with k+2 gives a square.

Original entry on oeis.org

7874, 8119, 69476962, 98010199, 108746354942, 449212110367, 544978035127, 870501316279, 998001001999, 1428394731903223, 1499870932756487, 1806498025502498, 1830668275445687, 1911470478658759, 2255786189655202
Offset: 1

Views

Author

Giovanni Resta, Jan 24 2006

Keywords

Comments

Numbers k such that k concatenated with k+1 gives the product of two numbers which differ by 2.
Numbers k such that k concatenated with k-2 gives the product of two numbers which differ by 4.
Numbers k such that k concatenated with k-7 gives the product of two numbers which differ by 6.

Examples

			8119//8121 = 9011^2, where // denotes concatenation.
98010199//98010200 = 99000100 * 99000102.
98010199//98010197 = 99000099 * 99000103.

Links

Crossrefs

Cf. A030465, A102567, A115427, A115428, A115429, A115430, A115431, A115432, A115433, A115434, A115435, A115436, A115437.
Cf. A116106, A116110, A116112, A116125, A116242.
Cf. A116136, A116143, A116107, A116275, A116163, A116099, A116177, A116295.

Programs

  • Python
    from itertools import count, islice
from sympy import sqrt_mod
def A115426_gen(): # generator of terms
    for j in count(0):
        b = 10**j
        a = b*10+1
        for k in sorted(sqrt_mod(2,a,all_roots=True)):
            if a*(b-2) <= k**2-2 < a*(a-3):
                yield (k**2-2)//a
A115426_list = list(islice(A115426_gen(),40)) # Chai Wah Wu, Feb 20 2024

Extensions

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

A115429 Numbers k such that the concatenation of k with k+8 gives a square.

Original entry on oeis.org

6001, 6433, 11085116, 44496481, 96040393, 115916930617, 227007035017, 274101929528, 434985419768, 749978863753, 996004003993, 1365379857457948, 1410590590957816, 1762388551055953, 2307340946901148, 2700383162251217
Offset: 1

Views

Author

Giovanni Resta, Jan 24 2006

Keywords

Comments

Also numbers k such that k concatenated with k+7 gives the product of two numbers which differ by 2.
Also numbers k such that k concatenated with k+4 gives the product of two numbers which differ by 4.
Also numbers k such that k concatenated with k-1 gives the product of two numbers which differ by 6.
Also numbers k such that k concatenated with k-8 gives the product of two numbers which differ by 8.

Examples

			6001//6009 = 7747^2, where // denotes concatenation.
96040393//96040400 = 98000200 * 98000202.
96040393//96040397 = 98000199 * 98000203.
96040393//96040392 = 98000198 * 98000204.

Crossrefs

Cf. A030465, A102567, A115426, A115437, A115428, A115430, A115431, A115432, A115433, A115434, A115435, A115436, A115440.
Cf. A116107, A116109, A116113, A116239.
Cf. A116205, A115430, A116335, A116185, A116187, A116317.
Cf. A116131, A116112, A116152, A116159, A116282.

Extensions

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

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

Original entry on oeis.org

6, 5346, 8083, 10578, 45531, 58626, 2392902, 2609443, 7272838, 51248898, 98009803, 159728062051, 360408196038, 523637103531, 770378933826, 998000998003, 1214959556998, 1434212848998, 3860012299771, 4243705560771
Offset: 1

Views

Author

Giovanni Resta, Jan 24 2006

Keywords

Comments

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)

Examples

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

Links

Crossrefs

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

Programs

  • Maple
    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

A115428 Numbers k such that the concatenation of k with k+5 gives a square.

Original entry on oeis.org

1, 4, 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

Comments

Also numbers k such that k concatenated with k+1 gives the product of two numbers which differ by 4.
Also numbers k such that k concatenated with k+4 gives the product of two numbers which differ by 2.

Examples

			14564_14569 = 38163^2.

Crossrefs

Cf. A030465, A102567, A115426, A115437, A115429, A115430, A115431, A115432, A115433, A115434, A115435, A115436, A115439.
Cf. A116125, A116099, A116167, A116297, A116177, A116171, A116185, A116315.

Extensions

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

A115430 Numbers k such that the concatenation of k with k+9 gives a square.

Original entry on oeis.org

216, 287, 515, 675, 1175, 4320, 82640, 960795, 1322312, 4049591, 16955015, 34602080, 171010235, 181964891, 183673467, 187160072, 321920055, 326530616, 328818032, 343942560, 470954312, 526023432, 528925616, 534830855
Offset: 1

Views

Author

Giovanni Resta, Jan 24 2006

Keywords

Comments

Also numbers k such that k concatenated with k+8 gives the product of two numbers which differ by 2.
Also numbers k such that k concatenated with k+5 gives the product of two numbers which differ by 4.

Examples

			82640_82649 = 90907^2.

Crossrefs

Cf. A030465, A102567, A115426, A115437, A115428, A115429, A115431, A115432, A115433, A115434, A115435, A115436, A115441.
Cf. A116172, A116168, A116344, A116112, A116193, A116173, A116323.

Extensions

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

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

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, Jan 25 2006

Keywords

Comments

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)

Examples

			9605_9601 = 9801^2.

Links

Crossrefs

Cf. A030465, A102567, A115426, A115437, A115428, A115429, A115430, A115431, A115433, A115434, A115435, A115436, A115443.
Almost the same as A116104 and A116121.

Programs

  • Maple
    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

A115435 Numbers k such that the concatenation of k with k-8 gives a square.

Original entry on oeis.org

2137, 2892, 6369, 12217, 21964, 28233, 42312, 4978977, 9571608, 18642249, 32288908, 96039609, 200037461217, 305526508312, 570666416233, 638912248204, 996003996009, 1846991026584, 3251664327537, 4859838227992
Offset: 1

Views

Author

Giovanni Resta, Jan 25 2006

Keywords

Examples

			18642249_18642241 = 43176671^2.

Crossrefs

Cf. A030465, A102567, A115426, A115437, A115428, A115429, A115430, A115431, A115432, A115433, A115434, A115436, A115446.

A115434 Numbers k such that the concatenation of k with k-7 gives a square.

Original entry on oeis.org

8, 16, 1337032, 2084503, 2953232, 4023943, 1330033613070195328, 4036108433661798551, 8283744867954114232, 6247320195351414276186411625291, 9452080202814205132771066881607
Offset: 1

Views

Author

Giovanni Resta, Jan 25 2006

Keywords

Examples

			4023943_4023936 = 6343456^2.

Crossrefs

Cf. A030465, A102567, A115426, A115437, A115428, A115429, A115430, A115431, A115432, A115433, A115435, A115436, A115445.

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 *)
Showing 1-10 of 11 results. Next

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

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