A116295 -id:A116295 - 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

A116286 Numbers k such that k*(k+2) gives the concatenation of a number m with itself.

Original entry on oeis.org

9, 99, 427, 572, 726, 845, 999, 7809, 9999, 36364, 63635, 99999, 326733, 673266, 999999, 4545453, 5454546, 9999999, 47058822, 52941177, 99999999, 331983806, 332667333, 384615385, 422892897, 475524476, 524475523, 577107102, 615384614, 667332666, 668016193, 719964245, 758241757, 804511279

Offset: 1

Giovanni Resta, Feb 06 2006

From Robert Israel, Apr 08 2025: (Start)
Numbers k such that k * (k + 2) = (10^d + 1) * m for some d and m where m has d digits.
Contains 10^d - 1 for all d >= 1. (End)

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A116136, A116279, A116285, A116287, A116295.

q:= proc(d,m) local R,t,a,b,x,q;
   t:= 10^d+1;
   R:= NULL;
   for a in numtheory:-divisors(t) do
     b:= t/a;
     if igcd(a,b) > 1 then next fi;
     for x from chrem([0,-m],[a,b]) by t do
       q:= x*(x+m)/t;
       if q >= 10^d then break fi;
       if q >= 10^(d-1) then R:= R, x fi;
   od od;
   sort(convert({R},list));
end proc:
A:=[seq(op(q(d,2)),d=1..10)]; # Robert Israel, Apr 08 2025

More terms from Robert Israel, Apr 08 2025

A116294 Numbers k such that k*(k+1) gives the concatenation of two numbers m and m+1.

Original entry on oeis.org

3, 7, 78, 80919, 91809, 326510, 475025, 524975, 673490, 4323777, 4767132, 5232868, 5676223, 4083911141, 4975000250, 5024999750, 5916088859, 9503960496, 9604950396, 4186904462792, 4313465946775, 5686534053225, 5813095537208, 7504715871407, 7631277355390

Offset: 1

Giovanni Resta, Feb 06 2006

			80919 * (80919 + 1) = 6547965480, the concatenation of 65479 and 65479 + 1.

Giovanni Resta, Table of n, a(n) for n = 1..2000

Cf. A116163, A116285, A116295, A116301.

Union @@ ((x /. List@ ToRules@ Reduce[x (x+1) == 10^# y +y+1 && x>0 && 10^(#-1) <= y+1 < 10^#, {x,y}, Integers]) & /@ Range[13] /. x->{}) (* Giovanni Resta, Jul 08 2018 *)

More terms from Giovanni Resta, Jul 08 2018

A116296 Numbers k such that k*(k+3) gives the concatenation of two numbers m and m+1.

Original entry on oeis.org

82, 77394228, 89158934, 36623663376237623663376336, 37633762366336633762366236, 62366237633663366237633762, 63376336623762376336623662, 86194223018927804587702129

Offset: 1

Giovanni Resta, Feb 06 2006

			89158934 * 89158937 = 79493157//79493158, where // denotes concatenation.

Cf. A116099, A116287, A116295, A116297, A116302.

A116308 Numbers k such that k*(k+2) is the concatenation of two numbers m and m+3.

Original entry on oeis.org

452, 547, 690, 855, 4381, 5618, 72729, 346532, 653467, 9090907, 94117645, 334665332, 336032386, 378253327, 390977441, 439928490, 483516485, 516483514, 560071509, 609022558, 621746672, 663967613, 665334667

Offset: 1

Giovanni Resta, Feb 06 2006

Cf. A116177, A116295, A116307, A116309, A116315.

cp's OEIS Frontend

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

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A116286 Numbers k such that k*(k+2) gives the concatenation of a number m with itself.

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A116294 Numbers k such that k*(k+1) gives the concatenation of two numbers m and m+1.

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A116296 Numbers k such that k*(k+3) gives the concatenation of two numbers m and m+1.

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A116308 Numbers k such that k*(k+2) is the concatenation of two numbers m and m+3.

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