A102567 -id:A102567 - cp's OEIS frontend

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

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.

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

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

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

A092118 Biperiod squares: square numbers whose digits repeat twice in order.

Original entry on oeis.org

1322314049613223140496, 2066115702520661157025, 2975206611629752066116, 4049586776940495867769, 5289256198452892561984, 6694214876166942148761, 8264462810082644628100, 183673469387755102041183673469387755102041

Offset: 1

Michael Mark, Dec 15 2004

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.

Author?, MMB message board "big square"
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. A102567, A106497.

f:=proc(n) local i,j,k; i:=cat(n,n); j:=convert(i,decimal,10); issqr(j); end;
with(numtheory): Digits:=50:for d from 1 to 22 do tendp1:=10^d+1: tendp1fact:=ifactors(tendp1)[2]: n:=mul(piecewise(tendp1fact[i][2] mod 2=1,tendp1fact[i][1],1),i=1..nops(tendp1fact)):for i from ceil(sqrt((10^(d-1))/n)) to floor(sqrt((10^d-1)/n)) do printf("%d, ",tendp1*n*i^2) od: od: # C. Ronaldo

from itertools import count, islice
from sympy import sqrt_mod
def A092118_gen(): # generator of terms
    for j in count(0):
        b = 10**j
        a = b*10+1
        ab, aa = a*b, a*(a-1)
        for k in sorted(sqrt_mod(0,a,all_roots=True)):
            if ab <= (m:=k**2) < aa:
                yield m
A092118_list = list(islice(A092118_gen(),10)) # Chai Wah Wu, Mar 06 2024

Corrected and extended by C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 15 2005
Definition corrected and improved, reference and cross-reference added by William Rex Marshall, Nov 12 2010
Keyword base added by William Rex Marshall, Nov 12 2010

A116279 Numbers k such that k*(k+2) gives the concatenation of two numbers m and m-1.

Original entry on oeis.org

36363636363, 45454545454, 54545454545, 63636363636, 72727272727, 81818181818, 90909090909, 428571428571428571428, 571428571428571428571, 714285714285714285714, 857142857142857142857

Offset: 1

Giovanni Resta, Feb 06 2006

			36363636363 * 36363636365 = 13223140496//13223140495, where // denotes concatenation.

Cf. A102567, A116273, A116280, A116286.

A115528 Numbers k such that k^2 is the concatenation of two numbers m and 2*m.

Original entry on oeis.org

6, 28571428571428571428572, 42857142857142857142858, 57142857142857142857144, 2247191011235955056179775280898876404494382022471910112360

Offset: 1

Giovanni Resta, Jan 25 2006

			6^2 = 3_6.

Cf. A102567, A106497, A115527 - A115556.

A115549 Numbers k such that the concatenation of k with 8*k gives a square.

Original entry on oeis.org

3, 12, 28, 63, 112, 278, 1112, 2778, 11112, 27778, 111112, 277778, 1111112, 2777778, 4938272, 7716050, 11111112, 12802888, 13151250, 13504288, 13862002, 14224392, 14591458, 14963200, 15339618, 15720712, 16106482, 16496928, 16892050, 17291848, 17696322, 18105472

Offset: 1

Giovanni Resta, Jan 25 2006

If k = 10*R_m + 2, with m >= 1, then the concatenation of k with 8*k equals (30*R_m + 6)^2, so A047855 \ {1,2} is a subsequence. - Bernard Schott, Apr 09 2022
Numbers k such that A009470(k) is a square. - Michel Marcus, Apr 09 2022
The numbers 28, 278, 2778, ..., 2*10^k + 7*(10^k - 1)/9 + 1, ..., k >= 1, are terms, because the concatenation forms the squares 28224 = 168^2, 2782224 = 1668^2, 277822224 = 16668^2, ..., (10^m + 2*(10^m - 1)/3 + 2)^2, m >= 2, ... - Marius A. Burtea, Apr 10 2022

			3_24 = 18^2.
11112_88896 = 33336^2.

Marius A. Burtea, Table of n, a(n) for n = 1..176

Cf. A002275, A047855, A009470, A102567, A106497, A115527-A115556.

[n:n in [1..20000000]|IsSquare(Seqint(Intseq(8*n) cat Intseq(n)))]; // Marius A. Burtea, Apr 10 2022

isok(k) = issquare(eval(Str(k, 8*k))); \\ Michel Marcus, Apr 09 2022

More terms from Marius A. Burtea, Apr 13 2022

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Crossrefs

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

Views

Author

Keywords

Examples

Crossrefs

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

Views

Author

Keywords

Examples

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A092118 Biperiod squares: square numbers whose digits repeat twice in order.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Python

Extensions

A116279 Numbers k such that k*(k+2) gives the concatenation of two numbers m and m-1.

Views

Author

Keywords

Examples

Crossrefs

A115528 Numbers k such that k^2 is the concatenation of two numbers m and 2*m.

Views

Author

Keywords

Examples

Crossrefs

A115549 Numbers k such that the concatenation of k with 8*k gives a square.

Views

Author

Keywords

Comments

Examples

Links