A092118 -id:A092118 - cp's OEIS frontend

A102567 Numbers k such that the concatenation of k with itself is a biperiod square.

13223140496, 20661157025, 29752066116, 40495867769, 52892561984, 66942148761, 82644628100, 183673469387755102041, 326530612244897959184, 510204081632653061225, 734693877551020408164

Offset: 1

C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 15 2005

Also, numbers N associated with A106497.
Also, numbers k such that k concatenated with k-1 gives the product of two numbers which differ by 2. E.g., 13223140496//13223140495 = 36363636363 * 36363636365, where // denotes concatenation. - Giovanni Resta and Franklin T. Adams-Watters, Nov 13 2006
From Jianing Song, Nov 01 2024: (Start)
Numbers 10^(k-1) <= a <= 10^k - 1 such that a*(10^k + 1) is a square. Note that 10^k + 1 must be nonsquarefree, i.e., k is in A086982, otherwise a must be divisible by 10^k + 1, which is impossible.
Let v(p,m) be the p-adic valuation of m.
 - If p is not in A045616, then v(p,10^k+1) = r > 0 if and only if v(p,gcd(n,10^k+1)) = r-1.
 - If p is in A045616, let e be the multiplicative order of 10 modulo p, then v(p,10^k+1) > 0 if and only if e is even and k is an odd multiple of e/2, in which case v(p,10^k+1) = v(p,10^e-1) + v(p,k) = v(p,10^e-1) + v(p,gcd(k,10^k+1)).
This helps to find the terms. (End)

			13223140496 concatenated with 13223140496 is 1322314049613223140496 = 36363636364^2.
40495867769 is in the sequence because writing it twice gives the square number 4049586776940495867769 = 63636363637^2.

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, A054214, A116163, A116136, A116279.

with(numtheory): Digits:=50:for d from 1 to 35 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, ",n*i^2) od: od:

A102567L[n_] := Catenate@Table[Module[{fac = FactorInteger[10^k + 1], min}, If[Max@fac[[All, -1]] == 1, {}, min = Times @@ Cases[fac, {a_, A102567L%5B30%5D%20(*%20_JungHwan%20Min">?OddQ} :> a]; Table[min s^2, {s, Ceiling@Sqrt[10^(k - 1)/min], Floor@Sqrt[(10^k - 1)/min]}]]], {k, n}]; A102567L[30] (* _JungHwan Min, Dec 11 2016 *)
A102567Q = IntegerQ@Sqrt@FromDigits[Join[#, #] &@IntegerDigits[#]] & (* JungHwan Min, Dec 11 2016 *)

p = [3, 487, 56598313]; \\ A045616
b(n) = my(d = gcd(n, lift(Mod(10,n)^n)+1), s = 1); for(j=1, #p, my(e = znorder(Mod(10, p[j]))); if((e % 2 == 0) && (n % (e/2) == 0) && (n/(e/2) % 2 == 1), my(v = valuation(d, p[j])); d /= p[j]^v; s *= p[j]^((v+valuation(10^e-1, p[j]))\2))); my(f = factor(d)); for(i=1, #f~, s *= f[i,1]^((f[i,2]+1)\2)); s; \\ giving s such that 10^n + 1 = s^2*t where t is squarefree, considering only the three already-known terms of A045616
A102567_length_n(n) = my(t = (10^n+1)/b(n)^2, lowlim = 1+sqrtint(10^(n-1)\t), uplim = sqrtint((10^n-1)\t)); vector(uplim-lowlim+1, i, (lowlim-1+i)^2 * t) \\ terms of the form a^2*t such that 10^(n-1) <= a^2*t <= 10^n - 1
\\ Jianing Song, Nov 01 2024

from itertools import count, islice
from sympy import sqrt_mod
def A102567_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**2//a
A102567_list = list(islice(A102567_gen(),10)) # Chai Wah Wu, Feb 19 2024

Entry revised by N. J. A. Sloane, Nov 14 2006 and also Nov 27 2006

Definition edited and reference added by William Rex Marshall, Nov 12 2010

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

Original entry on oeis.org

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

A181789 Pandigital biperiod squares: pandigital squares whose digits repeat twice in order.

Original entry on oeis.org

183673469387755102041183673469387755102041, 326530612244897959184326530612244897959184, 734693877551020408164734693877551020408164, 132231404958677685950413223140496132231404958677685950413223140496

Offset: 1

William Rex Marshall, Nov 12 2010

Ondrejka asks in Problem 1130(b) (see reference) what the smallest biperiod square is in which the ten decimal digits occur equally often (an equipandigital biperiod square), but it remains unknown whether any such square even exists.

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.

Chai Wah Wu, Table of n, a(n) for n = 1..34

Cf. A092118 (biperiod squares), A181790, A181791.

from itertools import count, islice
from sympy import sqrt_mod
def A181789_gen(): # generator of terms
    for j in count(9):
        b = 10**j
        a = b*10+1
        for k in sorted(sqrt_mod(0,a,all_roots=True)):
            if a*b <= (m:=k**2) < a*(a-1) and len(set(str(m//a))) == 10:
                    yield m
A181789_list = list(islice(A181789_gen(),20)) # Chai Wah Wu, Mar 23 2024

A381917 Kaprekar numbers that are the concatenation of two equal numbers.

Original entry on oeis.org

55, 99, 5050, 7272, 7777, 9999, 500500, 648648, 851851, 999999, 13641364, 24752475, 25252525, 36363636, 50005000, 61116111, 88888888, 99999999, 1111111111, 3888938889, 4132841328, 5000050000, 5243952439, 9756097560, 9999999999, 159341159341, 175676175676, 233415233415

Offset: 1

Shyam Sunder Gupta, Mar 10 2025

All repdigit numbers of even length which are Kaprekar numbers are terms. Since 99, 9999, 999999, ... and 5050, 500500, 50005000, ... are Kaprekar numbers, there are infinitely many terms.

			648648 is a Kaprekar number which is the concatenation of 648 and 648.

Shyam Sunder Gupta, Table of n, a(n) for n = 1..2163
Shyam Sunder Gupta, On Some Marvellous Numbers of Kaprekar, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315.

Intersection of A006886 and A020338.

Cf. A092118.

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A102567 Numbers k such that the concatenation of k with itself is a biperiod square.

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

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A181789 Pandigital biperiod squares: pandigital squares whose digits repeat twice in order.

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Author

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References

Links

Crossrefs

Programs

Python

A381917 Kaprekar numbers that are the concatenation of two equal numbers.

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