A115426 -id:A115426 - cp's OEIS frontend

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

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

Offset: 1

Giovanni Resta, Jan 25 2006

			4023943_4023936 = 6343456^2.

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

Giovanni Resta, Jan 25 2006

			638370_638361 = 798981^2.

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

Giovanni Resta, Jan 24 2006

From Farideh Firoozbakht, Nov 26 2006: (Start)
a(n).(a(n)+4) = A115438^2 where "." denotes concatenation.
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.
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)

			Using "." to denote concatenation, 120085.120089 = 346533^2.

Cf. A030465, A102567, A115426, A115428, A115429, A115430, A115431, A115432, A115433, A115434, A115435, A115436, A115438.

Select[Range[10^5],IntegerQ@Sqrt@FromDigits@Flatten[IntegerDigits/@{#,#+4}]&] (* Giorgos Kalogeropoulos, Jul 27 2021 *)

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

Giovanni Resta, Feb 06 2006

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

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

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A116129, A115431, A116099, A115426, A116267.

Cf. also A102567, A116154, A116130, A116286.

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

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

a(29)-a(32) from Chai Wah Wu, Feb 19 2024

A116125 Numbers k such that k concatenated with k-5 gives the product of two numbers which differ by 6.

Original entry on oeis.org

7, 97, 205, 300, 477, 732, 997, 1920, 3157, 9997, 52896, 99997, 120085, 427020, 999997, 8264460, 9999997, 88581312, 99999997, 112000885, 112917765, 143075580, 152863360, 193537077, 233788192, 266755221, 313680096

Offset: 1

Giovanni Resta, Feb 06 2006

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

			6786111717//6786111712 = 8237785936 * 8237785942, where // denotes concatenation.

Cf. A115426, A116124, A116126, A116131, A116256.

Cf. A116130, A116158, A115428, A116288.

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

A116107 Numbers k such that k concatenated with k-8 gives the product of two numbers which differ by 7.

Original entry on oeis.org

52, 63716866, 48793687600063875363014809897052, 60020753655608135708762056127156, 60446518621981165303188950156776, 71135436903815748345367595855336, 72876856643103028189103298533248

Offset: 1

Giovanni Resta, Feb 06 2006

Also numbers n such that n concatenated with n-2 gives the product of two numbers which differ by 5.

			63716866//63716864 = 79822843 * 79822848, where // denotes concatenation.

Cf. A116099, A116106, A115429, A116112, A116238, A115426, A116146, A116276.

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

A116163 Numbers k such that k concatenated with k+1 gives the product of two numbers which differ by 1.

Original entry on oeis.org

1, 5, 61, 65479, 84289, 106609, 225649, 275599, 453589, 1869505, 2272555, 2738291, 3221951, 1667833021, 2475062749, 2525062249, 3500010739, 9032526511, 9225507211, 1753016898055, 1860598847399, 3233666953849, 3379207972471, 5632076031055, 5823639407489

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 3.

			1 is a member since 12 = 3*4; also 10 = 2*5.
5 is a member since 56 = 7*8; also 54 = 6*9.

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

Cf. A116154, A115426, A116170, A116294.

Cf. A116143, A102567, A116112, A116130, A116280.

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

Edited by N. J. A. Sloane, Apr 15 2007, Jun 27 2009

More terms from Giovanni Resta, Jul 08 2018

A116275 Numbers k such that k*(k+4) gives the concatenation of two numbers m and m-2.

Original entry on oeis.org

8872, 9009, 83352840, 99000099, 329767122286, 670232877711, 738226276371, 933006600339, 999000000999, 3779410975143113, 3872816717528065, 4250291784692548, 4278630943941865, 4372036686326817, 4749511753491300, 5250488246508697, 5627963313673180, 5721369056058132

Offset: 1

Giovanni Resta, Feb 06 2006

			99000099 * 99000103 = 98010199//98010197, where // denotes concatenation.

Chai Wah Wu, Table of n, a(n) for n = 1..1187

Cf. A115426, A116267, A116274, A116276, A116288.

from itertools import count, islice
from sympy import sqrt_mod
def A116275_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+3) <= k**2-2 < a*(a+2):
                yield k-2
A116275_list = list(islice(A116275_gen(),30)) # Chai Wah Wu, Feb 19 2024

a(16)-a(18) from Chai Wah Wu, Feb 19 2024

A116295 Numbers k such that k*(k+2) gives the concatenation of two numbers m and m+1.

Original entry on oeis.org

8873, 9010, 83352841, 99000100, 329767122287, 670232877712, 738226276372, 933006600340, 999000001000, 3779410975143114, 3872816717528066, 4250291784692549, 4278630943941866, 4372036686326818, 4749511753491301

Offset: 1

Giovanni Resta, Feb 06 2006

From Robert Israel, Jun 06 2018: (Start)
Numbers k such that 10^m+1 | (k+1)^2-2 where (k+1)^2 has 2*m digits.
Includes 10^i - 10^(3*i) + 10^(4*i) for all i >= 1. (End)

			99000100 * 99000102 = 98010199//98010200, where // denotes concatenation.

Robert Israel, Table of n, a(n) for n = 1..600

Cf. A115426, A116286, A116294, A116296, A116308.

Res:= NULL:
for d from 1 to 40 do
  Res:= Res, op(sort(select(t -> t^2 >= 10^(2*d-1),map(t -> rhs(op(t))-1,[msolve(x^2=2, 10^d+1)]))))
od:
Res; # Robert Israel, Jun 06 2018

A116106 Numbers k such that k concatenated with k-8 gives the product of two numbers which differ by 6.

Original entry on oeis.org

9, 13, 99, 103, 183, 328, 528, 715, 999, 1003, 6099, 9999, 10003, 13224, 40495, 99999, 100003, 106755, 453288, 999999, 1000003, 2066115, 2975208, 9999999, 10000003, 22145328, 28027683, 99999999, 100000003, 110213248, 110667555

Offset: 1

Giovanni Resta, Feb 06 2006

			999999999//999999991 = 999999997 * 1000000003, where // denotes concatenation.

Cf. A116098, A116105, A116107, A115426, A116237.

cp's OEIS Frontend

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

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

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A115437 Numbers m such that the concatenation of m with m+4 gives a square.

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

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A116125 Numbers k such that k concatenated with k-5 gives the product of two numbers which differ by 6.

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A116107 Numbers k such that k concatenated with k-8 gives the product of two numbers which differ by 7.

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A116163 Numbers k such that k concatenated with k+1 gives the product of two numbers which differ by 1.

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A116275 Numbers k such that k*(k+4) gives the concatenation of two numbers m and m-2.

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A116295 Numbers k such that k*(k+2) gives the concatenation of two numbers m and m+1.

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Maple