A030465 -id:A030465 - cp's OEIS frontend

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

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

Offset: 1

Giovanni Resta, Jan 24 2006

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.

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

Chai Wah Wu, Table of n, a(n) for n = 1..1000

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.

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

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

Giovanni Resta, Jan 24 2006

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.

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

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.

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

A030467 Numbers k such that k^2 is a concatenation of two successive numbers.

Original entry on oeis.org

428, 573, 727, 846, 7810, 36365, 63636, 326734, 673267, 4545454, 5454547, 47058823, 52941178, 331983807, 332667334, 384615386, 422892898, 475524477, 524475524, 577107103, 615384615, 667332667, 668016194, 719964246, 758241758, 804511280, 810873337, 857142859

Offset: 1

Patrick De Geest

			428^2 = 183184, the concatenation of 183 and 184.

Chai Wah Wu, Table of n, a(n) for n = 1..10471 (terms n = 1..53 from Donovan Johnson, terms n = 54..1000 from Giovanni Resta)
Pante Stanica, Squares as concatenation of consecutive integers, Slides, West Coast Number Theory, Dec 17 2017.

Cf. A030465, A030466, A054214, A054215, A054216, A020339, A020340.

t={}; Do[If[EvenQ[y=Length[x=IntegerDigits[n^2]]] && Differences[FromDigits/@Partition[x,y/2]]=={1},AppendTo[t,n]],{n, 5.5*10^6}]; t (* Jayanta Basu, May 25 2013 *)
Sqrt[#]&/@(Select[FromDigits[Flatten[IntegerDigits/@#]]&/@ (Partition[ Range[735*10^6],2,1]),IntegerQ[Sqrt[#]]&]) (* The program takes a long time to run. *) (* Harvey P. Dale, Oct 10 2017 *)

for(n=1, 10^9, t=eval(concat(Str(n),Str(n+1))); if(issquare(t,&s), print1(s,", "))); /* Antonio Roldán and Joerg Arndt, Dec 31 2012 */

a(17) corrected by Donovan Johnson, Jan 03 2013

A054214 Numbers n such that n concatenated with n-1 is a square.

Original entry on oeis.org

82, 8242, 9802, 538277, 998002, 77837026, 99980002, 7922547265, 8643251345, 9223797610, 9999800002, 106710893290, 453378226757, 491023832065, 945958034530, 999998000002, 11916002265170, 15790977390245, 24917378001937, 25082758752026, 36315251812570

Offset: 1

Patrick De Geest, Feb 15 2000

Also, numbers k such that k concatenated with k-2 gives the product of two numbers which differ by 2.
Also, numbers n such that n concatenated with n-5 gives the product of two numbers which differ by 4.
Every term contains an even number of digits. - Max Alekseyev, May 14 2007
If n=(10^m-1)^2+1 where m is a positive integer then n is in the sequence. Because then n has 2m digits and n concatenated with n-1 is n*10^(2m)+(n-1) = (10^(2m)-10^m+1)^2. for example, taking m=1 we get 82, the first term of the sequence. - Farideh Firoozbakht, Aug 22 2013
As pointed out by Georg Fischer, it is very plausible that all of A054214, A116123, and A116142 contain exactly the same terms. If that is indeed true, then A054214 should be edited to mention the alternative constructions, and the other two sequences declared "dead". However, this needs careful analysis to deal with the possibilities that n, n-2, and n-5 may not all have the same number of digits. - N. J. A. Sloane, Oct 30 2018. Nov 05 2018: Thanks to Giovanni Resta_ (see below), this has now been done. - N. J. A. Sloane, Nov 05 2018
From Giovanni Resta, Nov 05 2018: (Start)
For n and n-5 to have a different digit length, we must have n = 10^k+h with 0<=h<=4.
We want to prove that in this case the concatenation of n and n-5 cannot be of the form m(m+4). The numbers m(m+4) modulo 9 can only be equal to 0, 3, 5, or 6, but it is easy to see that the concatenation of 10^k+h and 10^k+h-5 can be equal to one of these values modulo 9 only if h=0.
Now, the concatenation of 10^k and 10^k-5 is equal to 3 modulo 4 for every k>1, but m(m+4) modulo 4 can only be equal to 0 or 1, so A116123 is indeed equal to this sequence.
Using an identical argument (with mods 9 and 4) we can prove that the concatenation of n and n-2, when n and n-2 have a different number of digits, cannot be equal to m(m+2) and so A116142 is equal to this sequence. (End)

			E.g. '8242' + '8242-1' gives 82428241 which is 9079^2.

Luca, Florian, and Pantelimon Stănică. "Perfect Squares as Concatenation of Consecutive Integers." The American Mathematical Monthly 126.8 (2019): 728-734.

Giovanni Resta, Table of n, a(n) for n = 1..4751

Cf. A054215, A054216, A030465, A030466, A030467, A020339, A020340.

More terms from Max Alekseyev, May 14 2007
a(20)-a(21) from Giovanni Resta, Nov 05 2018
Edited by N. J. A. Sloane, Nov 05 2018

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

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

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

Giovanni Resta, Feb 06 2006

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.

			14564_14569 = 38163^2.

Cf. A030465, A102567, A115426, A115437, A115429, A115430, A115431, A115432, A115433, A115434, A115435, A115436, A115439.

Cf. A116125, A116099, A116167, A116297, A116177, A116171, A116185, A116315.

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

Giovanni Resta, Jan 24 2006

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.

			82640_82649 = 90907^2.

Cf. A030465, A102567, A115426, A115437, A115428, A115429, A115431, A115432, A115433, A115434, A115435, A115436, A115441.

Cf. A116172, A116168, A116344, A116112, A116193, A116173, A116323.

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

Giovanni Resta, Jan 25 2006

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)

			9605_9601 = 9801^2.

Robert Israel, Table of n, a(n) for n = 1..2805

Cf. A030465, A102567, A115426, A115437, A115428, A115429, A115430, A115431, A115433, A115434, A115435, A115436, A115443.

Almost the same as A116104 and A116121.

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

Giovanni Resta, Jan 25 2006

			18642249_18642241 = 43176671^2.

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

A115433 Numbers k such that the concatenation of k with k-5 gives a square.

Original entry on oeis.org

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

Offset: 1

Giovanni Resta, Jan 25 2006

			902406_902401 = 949951^2.

Cf. A030465, A102567, A115426, A115437, A115428, A115429, A115430, A115431, A115432, A115434, A115435, A115436, A115444.

cp's OEIS Frontend

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

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

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A030467 Numbers k such that k^2 is a concatenation of two successive numbers.

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A054214 Numbers n such that n concatenated with n-1 is a square.

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

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

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

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

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

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