A030465 -id:A030465 - 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 *)

A030466 Squares that are concatenations of two consecutive nonzero numbers.

Original entry on oeis.org

183184, 328329, 528529, 715716, 60996100, 1322413225, 4049540496, 106755106756, 453288453289, 20661152066116, 29752082975209, 2214532822145329, 2802768328027684, 110213248110213249, 110667555110667556, 147928995147928996, 178838403178838404, 226123528226123529

Offset: 1

Patrick De Geest

British Mathematical Olympiad, 1993, Round 1, Question 1: "Find, showing your method, a six-digit integer n with the following properties: (i) n is a perfect square, (ii) the number formed by the last three digits of n is exactly one greater than the number formed by the first three digits of n. (Thus n might look like 123124, although this is not a square.)"
Steve Dinh, The Hard Mathematical Olympiad Problems And Their Solutions, AuthorHouse, 2011, Problem 1 of the British Mathematical Olympiad 1993, page 164.

Iain Fox, Table of n, a(n) for n = 1..10471.
British Mathematical Olympiad 1993, Round 1, Problem 1.
Michael Penn, British Mathematics Olympiad 1993 Round 1 Question 1, YouTube, Apr 24, 2020.
Pante Stanica, Squares as concatenation of consecutive integers, Slides, West Coast Number Theory, Dec 17 2017.
Index to sequences related to Olympiads.

Cf. A000993, A030465, A030467, A054214, A054215, A054216, A020339, A020340.

fQ[n_] := IntegerQ[Sqrt[n*10^Floor[1 + Log10[n + 1]] + n + 1]]; (* Robert G. Wilson v, Dec 27 2017 *)

lista(nn) = forstep(n=183, nn, [3, 5, 7, 5, 3, 1, 4, 7, 5, 3, 5, 7, 5, 3, 5, 7, 5, 3, 5, 7, 4, 1], my(s = eval(concat(Str(n), Str(n+1)))); if(issquare(s), print1(s, ", "))) \\ Iain Fox, Dec 27 2017

eea(x, y) = my(a=max(x,y), b=min(x,y), s=0, so=1, st, r=b, ro=a, rt, q, t); while(r, q=ro\r; rt=r; r=ro-q*r; ro=rt; st=s; s=so-q*s; so=st); t=(ro-so*a)\b; if(x>y, [so, t], [t, so]) \\ Extended Euclidean Algorithm
lista(nn) = my(res=Set(), b, f2, c, s); for(d=3, nn, b=10^d+1; fordiv(b, f, if(f!=1 && f!=b, f2=b/f; if(gcd(f, f2)==1, c=eea(f, f2); if(c[1]<0, s=f*(f2+2*c[1])*f2*(f-2*c[2])+1, s=f*(2*c[1])*f2*(-2*c[2])+1); if(#digits(s)==d*2, res=setunion(res, Set(s))))))); Vec(res) \\ (Will find all values of length nn*2 or shorter) Iain Fox, Oct 16 2021

a(n) = A030465(n)*(10^A055642(A030465(n))+1)+1. - Iain Fox, Oct 16 2021

a(15)-a(17) from Arkadiusz Wesolowski, Apr 02 2014

a(18) from Iain Fox, Dec 27 2017

A054216 Numbers m such that m^2 is a concatenation of two consecutive decreasing numbers.

Original entry on oeis.org

91, 9079, 9901, 733674, 999001, 88225295, 99990001, 8900869208, 9296908812, 9604060397, 9999900001, 326666333267, 673333666734, 700730927008, 972603739727, 999999000001, 34519562953737, 39737862788838, 49917309624956

Offset: 1

Patrick De Geest, Feb 15 2000

Obviously b(n) = 100^n - 10^n + 1 = (91, 9901, 999001, 99990001, ...) is a subsequence. Are { b(2), b(4), b(6), b(8) } the only terms of this sequence that are prime? - M. F. Hasler, Mar 30 2008. Answer: The smallest prime in this sequence that is not of the form b(n) is A054216(155) = 811451682377384625400019885321 [Max Alekseyev, Oct 08 2008]. See A145381 for further prime terms.
Other subsequences are c(n) = ( 10^(6n) - 2*10^(5n) - 10^(3n) - 2*10^n + 1 )/3 (n>=2), d(n) = (33/101)*(100^(404n+71)+1)+10^(404n+71) (n>=0) and e(n) = (33/101)*(100^(404n-71)+1)+10^(404n-71) (n>=1). Primes among these include c(10), c(14) and d(0). - M. F. Hasler, Oct 09 2008
A positive integer m is in this sequence if and only if m^2 == -1 (mod 10^k + 1) where k is the number of decimal digits in m. Note that k cannot be odd, since in this case 11 divides 10^k + 1 while -1 is not a square modulo 11. - Max Alekseyev, Oct 09 2008

			'8242' + '8242-1' gives 82428241 which is 9079^2.
Leading zeros are not allowed, which is why c(1)=266327 is not in this sequence although c(1)^2 = 070930 070929.

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..1000

Cf. A020339, A020340, A030465, A030466, A030467, A054214, A054215, A145381.

isA054216(n)={ 1==[1,-1]*divrem(n^2,10^(#Str(n^2)\2)) & #Str(n^2)%2==0 }

a(n) = sqrt(A054215(n)). - Max Alekseyev, May 14 2007

More terms from Max Alekseyev, May 14 2007

Several corrections and additions from M. F. Hasler, Oct 09 2008

A054215 Squares that are concatenations of two consecutive decreasing numbers.

Original entry on oeis.org

8281, 82428241, 98029801, 538277538276, 998002998001, 7783702677837025, 9998000299980001, 79225472657922547264, 86432513458643251344, 92237976109223797609, 99998000029999800001, 106710893290106710893289

Offset: 1

Patrick De Geest, Feb 15 2000

Infinitely many terms of this sequence are provided by A168624(k)^2 for k>0. - Bruno Berselli, Mar 13 2018

			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.

Cf. A054214, A054216, A030465, A030466, A030467, A020339, A020340.

a(n) = concatenation of A054214(n) and A054214(n)-1. - Max Alekseyev, May 14 2007

More terms from Max Alekseyev, May 14 2007

82848241 corrected to 82428241 by Dominick Cancilla, Jul 21 2010

A020339 a(n)^2 is the least square base-n doublet (base-n representation is the concatenation of 2 identical strings).

Original entry on oeis.org

6, 2, 615, 84, 119973, 4, 3, 23620, 36363636364, 6, 24766945690, 17928148, 915, 4, 86808207405692007605, 6, 130, 10, 2667, 95530227420606, 10623969116570, 12, 5, 343872950627253606, 9, 14, 59239353339085, 8130

Offset: 2

David W. Wilson

The identical strings must contain at least one nonzero digit, so that a(n) > 0. - Alonso del Arte, Jun 20 2018

In Bridy et al. it is shown how to construct an example (although not necessarily the least example) for each integer base n >= 2. - Jeffrey Shallit, Jun 14 2021

			The first few squares in binary are 1, 100, 1001, 10000, 11001, 100100. Thus we see that 100100, which is 36 in decimal, the square of 6, is the first square which is the concatenation of two identical bit patterns, and therefore a(2) = 6.

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.
David Wells, "The Penguin Dictionary of Curious and Interesting Numbers", Revised Edition 1997, p. 189.

Robert Israel, Table of n, a(n) for n = 2..99
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.
A. Ottens, The arithmetic-digits-squares-three.digits problem

Cf. A020340, A054214, A054215, A054216, A030465, A030466, A030467.

f:= proc(b)
  local d,F,x,t,j;
  for d from 1 do
    F:= select(t -> t[2]::odd, ifactors(1+b^d)[2]);
    x:= mul(t[1],t=F);
    if x >= b^d then next fi;
    j:= ceil(sqrt(b^(d-1)/x));
    if j^2*x < b^d then return j*sqrt(x*(1+b^d)) fi
  od
end proc:
map(f, [$2..40]); # Robert Israel, May 19 2024

a(j*k^2-1) = j if k >= 2 and j is squarefree. - Robert Israel, May 19 2024

Name slightly adjusted by Alonso del Arte, Jun 20 2018

A020340 Least square base n doublet (written in base 10).

Original entry on oeis.org

36, 4, 378225, 7056, 14393520729, 16, 9, 557904400, 1322314049613223140496, 36, 613401598811409576100, 321418490709904, 837225, 16, 7535664872989640713426833504575377836025, 36, 16900, 100, 7112889

Offset: 2

David W. Wilson

In Bridy et al. it is shown how to construct infinitely many examples for any given base n >= 2. - Jeffrey Shallit, Jun 14 2021

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.
David Wells, "The Penguin Dictionary of Curious and Interesting Numbers", Revised Edition 1997, p. 189.

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.
A. Ottens, The arithmetic-digits-squares-three.digits problem

Cf. A020339, A054214, A054215, A054216, A030465, A030466, A030467.

A233358 Numbers n such that n^2 concatenated with n + 1 is prime.

Original entry on oeis.org

2, 6, 30, 32, 38, 48, 50, 72, 90, 92, 98, 102, 110, 126, 128, 150, 182, 188, 212, 218, 230, 242, 258, 300, 318, 348, 398, 420, 440, 450, 462, 498, 516, 518, 530, 542, 572, 582, 636, 660, 662, 678, 702, 716, 726, 728, 746, 786, 798, 830, 848, 872, 878, 890, 908

Offset: 1

K. D. Bajpai, Dec 07 2013

All the terms in sequence are even numbers, but none are congruent to 4 mod 10.

			6 is in the sequence because the concatenation of 6^2 and 6 + 1 gives 367, which is prime.
30 is in the sequence because the concatenation of 30^2 and 31 gives 90031, which is prime.
34 is not in the sequence because the concatenation of 34^2 and 35 gives 115635 = 3 * 5 * 13 * 593.

K. D. Bajpai, Table of n, a(n) for n = 1..5550

Cf. A030465 (numbers n: concatenate n with n + 1 is prime).

KD := proc() local a,b; a:=parse(cat(n^2,n+1)); if  isprime(a) then RETURN (n); fi; end: seq(KD(), n=1..2000);

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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A030466 Squares that are concatenations of two consecutive nonzero numbers.

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PARI

PARI

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A054216 Numbers m such that m^2 is a concatenation of two consecutive decreasing numbers.

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PARI

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A054215 Squares that are concatenations of two consecutive decreasing numbers.

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A020339 a(n)^2 is the least square base-n doublet (base-n representation is the concatenation of 2 identical strings).

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Maple

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A020340 Least square base n doublet (written in base 10).

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A233358 Numbers n such that n^2 concatenated with n + 1 is prime.