A102567 -id:A102567 - cp's OEIS frontend

A181790 Numbers k such that k concatenated with itself is a pandigital biperiod square.

183673469387755102041, 326530612244897959184, 734693877551020408164, 132231404958677685950413223140496, 206611570247933884297520661157025, 297520661157024793388429752066116, 404958677685950413223140495867769

Offset: 1

William Rex Marshall, Nov 12 2010

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. A102567, A181789, A181791.

from itertools import count, islice
from sympy import sqrt_mod
def A181790_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 <= k**2 < a*(a-1) and len(set(str(m:=k**2//a))) == 10:
                    yield m
A181790_list = list(islice(A181790_gen(),20)) # Chai Wah Wu, Mar 23 2024

A115529 Numbers k such that the concatenation of 2*k with k gives a square.

Original entry on oeis.org

1632653061224489796, 3673469387755102041, 6530612244897959184, 108166576527852893455922120064900, 130881557598702001081665765278529, 155759870200108166576527852893456, 182801514332071389940508382909681

Offset: 1

Giovanni Resta, Jan 25 2006

Numbers of the form k = a*b^2 where 10^(d-1) <= k < 10^d and (2*10^d+1)/a is a square. - Robert Israel, Jan 13 2021

			3265306122448979592_1632653061224489796 = 5714285714285714286^2.

Robert Israel, Table of n, a(n) for n = 1..36

Cf. A102567, A106497, A115527 - A115556.

f:= proc(d) local R,q,F,G,s,t,a,u,i;
   q:= 2*10^d+1;
   F:= ifactors(q)[2];
   G:= map(t -> [t[1],floor(t[2]/2)], F);
   s:= mul(t[1]^t[2],t=G);
   R:= NULL:
   for a in numtheory:-divisors(s) do
     u:= q/a^2;
     R:= R, seq(i^2*u, i=ceil(sqrt(10^(d-1)/u))..floor(sqrt((10^d-1)/u)))
   od;
   R
end proc:
seq(f(d),d=1..33); # Robert Israel, Jan 13 2021

A115531 Numbers k such that the concatenation of k with 3*k gives a square.

Original entry on oeis.org

816326530612244897959183673469388, 1836734693877551020408163265306123, 3265306122448979591836734693877552, 3746097814776274713839750260145681581685744016649323621228

Offset: 1

Giovanni Resta, Jan 25 2006

If 3+10^m is not squarefree, say 3+10^m = u^2*v where v is squarefree, then the terms with length m are t^2*v where 10^m > 3*t^2*v >= 10^(m-1). The first m for which 3+10^m is not squarefree are 34, 59, 60, 61, 67. - Robert Israel, Aug 07 2019

Since 3+10^m is divisible by 7^2 for m = 34 + 42*k, the sequence contains 4*(3+10^m)/49, 9*(3+10^m)/49 and 16*(3+10^m)/49 for such m, and in particular is infinite. - Robert Israel, Aug 08 2019

Robert Israel, Table of n, a(n) for n = 1..113

Cf. A102567, A106497, A115527 - A115556.

Res:= NULL:
for m from 1 to 67 do
if not numtheory:-issqrfree(3+10^m) then
   F:= select(t -> t[2]=1, ifactors(3+10^m)[2]);
   v:= mul(t[1], t=F);
   Res:= Res, seq(t^2*v, t = ceil(sqrt(10^(m-1)/(3*v))) .. floor(sqrt(10^m/(3*v))))
fi
od:
Res;  # Robert Israel, Aug 07 2019

A115555 Numbers k such that the concatenation of 9*k with k gives a square.

Original entry on oeis.org

1836734693877551020408163265306122449, 7346938775510204081632653061224489796, 15311909262759924385633270321361058601134215500945179584121, 27221172022684310018903591682419659735349716446124763705104

Offset: 1

Giovanni Resta, Jan 25 2006

Cf. A102567, A106497, A115527 - A115556.

A107678 Numbers NM associated with A107677.

Original entry on oeis.org

74, 8021, 9704, 165836, 352649, 997004, 19768025, 20657936, 29767025, 30856916, 98000201, 99970004, 1203287969, 1975280249, 3086569136, 4265757344, 8143718564, 9999700004, 48125951876, 81632918369, 82205917796, 87792912209

Offset: 1

Lekraj Beedassy, May 20 2005

Leading zeros are omitted.

Cf. A107677, A102567.

Edited, corrected and extended by Klaus Brockhaus, May 22 2005

A115530 Numbers k such that k^2 is the concatenation of two numbers 2*m and m.

Original entry on oeis.org

5714285714285714286, 8571428571428571429, 11428571428571428572, 465116279069767441860465116279070, 511627906976744186046511627906977, 558139534883720930232558139534884, 604651162790697674418604651162791

Offset: 1

Giovanni Resta, Jan 25 2006

			5714285714285714286^2 = 3265306122448979592_1632653061224489796.

Cf. A102567, A106497, A115527 - A115556.

Definition modified by Georg Fischer, Jul 25 2019

A115532 Numbers k such that k^2 is the concatenation of two numbers m and 3*m.

Original entry on oeis.org

2857142857142857142857142857142858, 4285714285714285714285714285714287, 5714285714285714285714285714285716, 19354838709677419354838709677419354838709677419354838709678

Offset: 1

Giovanni Resta, Jan 25 2006

Cf. A102567, A106497, A115527 - A115556.

A115533 Numbers k such that the concatenation of 3*k with k gives a square.

Original entry on oeis.org

24489796, 55102041, 97959184, 15976331361, 28402366864, 44378698225, 63905325444, 86982248521, 24489795918367346938775510204081632653061224489796

Offset: 1

Giovanni Resta, Jan 25 2006

			73469388_24489796 = 85714286^2.

Cf. A102567, A106497, A115527 - A115556.

A115534 Numbers k such that k^2 is the concatenation of two numbers 3*m and m.

Original entry on oeis.org

85714286, 128571429, 171428572, 69230769231, 92307692308, 115384615385, 138461538462, 161538461539, 85714285714285714285714285714285714285714285714286

Offset: 1

Giovanni Resta, Jan 25 2006

			85714286^2 = 73469388_24489796.

Cf. A102567, A106497, A115527 - A115556.

Definition modified by Georg Fischer, Jul 26 2019

A115535 Numbers k such that the concatenation of k with 4*k gives a square.

Original entry on oeis.org

25721, 28836, 32129, 35600, 39249, 43076, 47081, 51264, 55625, 60164, 64881, 69776, 74849, 80100, 85529, 91136, 96921, 102884, 109025, 115344, 121841, 128516, 135369, 142400, 149609, 156996, 164561, 172304, 180225, 188324, 196601, 205056

Offset: 1

Giovanni Resta, Jan 25 2006

Not the same as P(n) = 89*n^2 + 2848*n + 22784: P(n) = a(n) for a < 37, but a(37) = 3698225. - Charles R Greathouse IV, Jul 28 2010

			25721_102884 = 160378^2.

Cf. A102567, A106497, A115527 - A115556.

Select[Range[250000],IntegerQ[Sqrt[#*10^IntegerLength[4*#]+4*#]]&] (* Harvey P. Dale, Jan 04 2019 *)

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A181790 Numbers k such that k concatenated with itself is a pandigital biperiod square.

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A115529 Numbers k such that the concatenation of 2*k with k gives a square.

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A115531 Numbers k such that the concatenation of k with 3*k gives a square.

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Maple

A115555 Numbers k such that the concatenation of 9*k with k gives a square.

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A107678 Numbers NM associated with A107677.

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A115530 Numbers k such that k^2 is the concatenation of two numbers 2*m and m.

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A115532 Numbers k such that k^2 is the concatenation of two numbers m and 3*m.

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A115533 Numbers k such that the concatenation of 3*k with k gives a square.

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A115534 Numbers k such that k^2 is the concatenation of two numbers 3*m and m.

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A115535 Numbers k such that the concatenation of k with 4*k gives a square.

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Mathematica