A115527 -id:A115527 - cp's OEIS frontend

A115556 Numbers whose square is the concatenation of two numbers 9*m and m.

12857142857142857142857142857142857143, 25714285714285714285714285714285714286, 117391304347826086956521739130434782608695652173913043478261

Offset: 1

Giovanni Resta, Jan 25 2006

a(4)=156521739130434782608695652173913043478260869565217391304348.
From Robert Israel, Aug 24 2023: (Start)
If 9 * 10^d + 1 = a^2 * b with a > 1, then a * b * c is a term if a^2/(90 + 10^(1-d)) < c^2 < a^2/(9 + 10^(-d)).  For example, 9 * 10^d + 1 is divisible by 7^2 for d == 37 (mod 42), and then (9 * 10^d + 1)/7 and 2*(9 * 10^d + 1)/7 are terms.  In particular, the sequence is infinite. (End)

Robert Israel, Table of n, a(n) for n = 1..12

Cf. A009474, A102567, A106497, A115527 - A115555.

F:= proc(d) local R,F,t,b,r,q,s,m0,x0,k;
     R:= NULL;
     F:= ifactors(9*10^d+1)[2];
     b:= mul(t[1]^floor(t[2]/2),t=F);
     for r in numtheory:-divisors(b) do
       x0:= (9*10^d+1)/r;
       m0:= x0/r;
       for k from ceil(sqrt(10^(d-1)/m0)) to floor(sqrt(10^d/m0)) do
         R:= R, x0*k;
       od
     od;
       R
end proc:
sort(map(F, [$1..90])); # Robert Israel, Aug 24 2023

Definition modified by Georg Fischer, Jul 26 2019

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

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.

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

cp's OEIS Frontend

A115556 Numbers whose square is the concatenation of two numbers 9*m and m.

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

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A115549 Numbers k such that the concatenation of k with 8*k gives a 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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A115555 Numbers k such that the concatenation of 9*k with k gives a square.

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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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