A272684 -id:A272684 - cp's OEIS frontend

A272685 a(n) = smallest m such that A265432(m) = A272684(n), or -1 if no such m exists.

1, 15, 14, 13, 12, 217, 0, 215, 45, 213, 44, 43, 209, 42, 207, 2, 573, 1327, 572, 130, 185, 570, 1492, 569, 78, 568, 128, 567, 1318, 1498, 565, 188, 564, 10, 563, 1312, 562, 1504, 1309, 560, 1507, 693, 74, 558, 1510, 557, 192, 2360, 6085, 350, 1480, 708, 6083, 3643

Offset: 1

Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 24 2016

It is conjectured that all terms of A272684 eventually appear in A265432.

See A273369 for a version that ignores the fact that terms in A272671 ending in 0 cannot appear in A265432.

			A272684(1) = 6 first appears in A265432 at index 1, so a(1) = 1.
A272684(7) = 225 first appears in A265432 at index 0, so a(7) = 0.

Cf. A272684, A272671, A265432.

See A273369 for another version.

A272671 Numbers k such that the decimal number 1k is a square.

Original entry on oeis.org

6, 21, 44, 69, 96, 156, 225, 296, 369, 444, 521, 600, 681, 764, 849, 936, 1025, 1236, 1449, 1664, 1881, 2100, 2321, 2544, 2769, 2996, 3225, 3456, 3689, 3924, 4161, 4400, 4641, 4884, 5129, 5376, 5625, 5876, 6129, 6384, 6641, 6900, 7161, 7424, 7689, 7956, 8225, 8496, 8769, 9044, 9321, 9600

Offset: 1

Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 20 2016

			44 is a member because 144 = 12^2 is a square.
0 is not a member because 10 is not a square.

Davin Park, Table of n, a(n) for n = 1..20000

Cf. A265432, A272672, A045855 (squares beginning with 1), A272684, A272685.

[n: n in [1..10000 ] | IsSquare(Seqint(Intseq(n) cat Intseq(1)))]; // Marius A. Burtea, Mar 21 2019

t1:=[];
for k from 1 to 20000 do
if issqr(k+10^length(k)) then t1:=[op(t1),k]; fi;
od;
t1;

Flatten[n /. Solve[10^# + n == a^2 && 10^(# - 1) <= n < 10^# && a > 0, {n, a}, Integers] & /@ Range[3]] (* Davin Park, Feb 05 2017 *)
Select[Range[10000],IntegerQ[Sqrt[10^IntegerLength[#]+#]]&] (* Harvey P. Dale, Jul 20 2025 *)

isok(n) = issquare(eval(concat(1, Str(n)))); \\ Michel Marcus, Mar 21 2019

from sympy.ntheory.primetest import is_square
def ok(n): return is_square(int('1'+str(n)))
print(list(filter(ok, range(9601)))) # Michael S. Branicky, Jun 21 2021

Extended by Davin Park, Feb 05 2017

A265432 a(n) = smallest k with concat(1,k) and concat(n,k) both square numbers.

Original entry on oeis.org

225, 6, 1025, 6, 225, 9937257544619140625, 80625, 225, 19025, 14797831640625, 5625, 89450791534674072265625, 96, 69, 44, 21, 1993672119140625, 2002541101386962890625, 225, 6, 8734765625, 99758030615478515625, 5625, 863225, 80625, 6, 40625, 225, 890625, 158764150390625

Offset: 0

David W. Wilson and Robert G. Wilson v, Dec 08 2015

k must be a positive integer (and of course cannot begin with 0). - N. J. A. Sloane, May 19 2016

Every term is a member of A272671, by definition. Certainly every term of A272671 which is a power of 100 times an earlier term of A272671 (such as 600, 2100, 4400) will not appear, by the "smallest k" condition. Does every other term of A272671 (that is, the terms of A272684) eventually appear? See A272685 and A273369 for the first appearance of these terms. - Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 23 2016

			a(0) = 225 because 1225 is a square as is (0)225. (In other words, 225 is the first term in A272672 that is itself a square). - _N. J. A. Sloane_, May 21 2016
a(2) = 1025 because concat(1,1025) = 11025 = 105^2 and concat(2,1025) = 21025 = 145^2.

N. J. A. Sloane, Table of n, a(n) for n = 0..20000, May 25 2016 [First 10000 terms from David W. Wilson, Dec 08 2015]
N. J. A. Sloane, Table of n, a(n), sqrt(concat(n,a(n))), sqrt(concat(1,a(n))) (based on _David W. Wilson_ 10000-term b-file)
David W. Wilson, Table of n, a(n) for n = 0..10000, May 25 2016 [The graph of the first 10000 terms looks rather different from the graph of the first 20000 terms, so this file is worth preserving. - _N. J. A. Sloane_, May 25 2016]

Cf. A045855, A272671, A018796, A272684, A272685 and A273369 (smallest inverse).
For records see A272674, A272675.
For square roots referred to in definition see A272682, A272683.
A018851 is a simpler sequence in the same spirit.

<< Combinatorica`
A265432[0] = 225;
A265432[1] = 6;
A265432[n_] := Block[{x = {-1, 1, 0, 1}[[Mod[n, 4, 1]]], d = Infinity, l, i}, While[d > Sqrt[10.0^(x - 1)] (Sqrt[10.0 n + 1] - Sqrt[11.0]), x++; d = Infinity; l = Divisors[((n - 1) 10^x)/4]; i = BinarySearch[l, 0.5 Sqrt[(n + 1) 10.0^x - 1] - 0.5 Sqrt[2*10.0^x - 1]]; If[i <= Length@l, d = 2*l[[i + 1/2]]]]; (((n - 1) 10^x - d^2)/(2 d))^2 - 10^x] (* Davin Park, Apr 11 2017 *)

cp's OEIS Frontend

A272685 a(n) = smallest m such that A265432(m) = A272684(n), or -1 if no such m exists.

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A272671 Numbers k such that the decimal number 1k is a square.

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A265432 a(n) = smallest k with concat(1,k) and concat(n,k) both square numbers.

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Mathematica