A102567 -id:A102567 - cp's OEIS frontend

A106497 Numbers whose square is the concatenation of two identical numbers, i.e., of the form NN.

36363636364, 45454545455, 54545454546, 63636363637, 72727272728, 81818181819, 90909090910, 428571428571428571429, 571428571428571428572, 714285714285714285715, 857142857142857142858

Offset: 1

Lekraj Beedassy, May 04 2005

For the corresponding numbers N see A102567.

Numbers of the form j*(10^d + 1)/k where 10^d + 1 == 0 (mod k^2) and k/sqrt(10) < j < k. - David W. Wilson, Nov 09 2006

			63636363637 is in the sequence because 63636363637^2 = 4049586776940495867769 is 40495867769 written twice.

Andrew Bridy, Robert J. Lemke Oliver, Arlo Shallit, and Jeffrey Shallit, The Generalized Nagell-Ljunggren Problem: Powers with Repetitive Representations, Experimental Math, 28 (2019), 428-439.
R. Ondrejka, Problem 1130: Biperiod Squares, Journal of Recreational Mathematics, Vol. 14:4 (1981-82), 299. Solution by F. H. Kierstead, Jr., JRM, Vol. 15:4 (1982-83), 311-312.

David W. Wilson, Table of n, a(n) for n = 1..1098
Dr Barker, Can Numbers Like These Be Square?, YouTube video, 2023.
Andrew Bridy, Robert J. Lemke Oliver, Arlo Shallit, and Jeffrey Shallit, The Generalized Nagell-Ljunggren Problem: Powers with Repetitive Representations, preprint arXiv:1707.03894 [math.NT], July 14 2017.

Cf. A092118, A102567.

from itertools import count, islice
from sympy import sqrt_mod
def A106497_gen(): # generator of terms
    for j in count(0):
        b = 10**j
        a = b*10+1
        for k in sorted(sqrt_mod(0,a,all_roots=True)):
            if a*b <= k**2 < a*(a-1):
                yield k
A106497_list = list(islice(A106497_gen(),10)) # Chai Wah Wu, Feb 19 2024

a(7) from Klaus Brockhaus, May 06 2005
More terms from David W. Wilson, Nov 05 2006
Reference and cross-references added by William Rex Marshall, Nov 12 2010

A115556 Numbers whose square is the concatenation of two numbers 9*m and m.

Original entry on oeis.org

12857142857142857142857142857142857143, 25714285714285714285714285714285714286, 117391304347826086956521739130434782608695652173913043478261

Offset: 1

Giovanni Resta, Jan 25 2006

a(4)=156521739130434782608695652173913043478260869565217391304348.
From Robert Israel, Aug 24 2023: (Start)
If 9 * 10^d + 1 = a^2 * b with a > 1, then a * b * c is a term if a^2/(90 + 10^(1-d)) < c^2 < a^2/(9 + 10^(-d)).  For example, 9 * 10^d + 1 is divisible by 7^2 for d == 37 (mod 42), and then (9 * 10^d + 1)/7 and 2*(9 * 10^d + 1)/7 are terms.  In particular, the sequence is infinite. (End)

Robert Israel, Table of n, a(n) for n = 1..12

Cf. A009474, A102567, A106497, A115527 - A115555.

F:= proc(d) local R,F,t,b,r,q,s,m0,x0,k;
     R:= NULL;
     F:= ifactors(9*10^d+1)[2];
     b:= mul(t[1]^floor(t[2]/2),t=F);
     for r in numtheory:-divisors(b) do
       x0:= (9*10^d+1)/r;
       m0:= x0/r;
       for k from ceil(sqrt(10^(d-1)/m0)) to floor(sqrt(10^d/m0)) do
         R:= R, x0*k;
       od
     od;
       R
end proc:
sort(map(F, [$1..90])); # Robert Israel, Aug 24 2023

Definition modified by Georg Fischer, Jul 26 2019

A115527 Numbers k such that the concatenation of k with 2*k gives a square.

Original entry on oeis.org

3, 8163265306122448979592, 18367346938775510204082, 32653061224489795918368, 504986744097967428355005681100871102133568993813912384800

Offset: 1

Giovanni Resta, Jan 25 2006

			3_6 = 6^2.

Cf. A102567, A106497, A115528 - A115556.

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

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

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.

cp's OEIS Frontend

A106497 Numbers whose square is the concatenation of two identical numbers, i.e., of the form NN.

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A115556 Numbers whose square is the concatenation of two numbers 9*m and m.

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A115527 Numbers k such that the concatenation of k with 2*k gives a square.

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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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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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