A045855 -id:A045855 - cp's OEIS frontend

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

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

A045784 Squares with initial digit '1'.

Original entry on oeis.org

1, 16, 100, 121, 144, 169, 196, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 10000, 10201, 10404, 10609, 10816, 11025, 11236, 11449, 11664, 11881, 12100, 12321, 12544, 12769, 12996, 13225, 13456, 13689, 13924

Offset: 1

Jeff Burch

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

Cf. A045855.

Cf. A045785, A045786, A045787, A045788, A045789, A045791, A045792, A045793.

Select[Range[120]^2,First[IntegerDigits[#]]==1&] (* Harvey P. Dale, Dec 31 2011 *)

a(n) = A045855(n)^2. - Michel Marcus, Sep 04 2021

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 *)

A348487 Positive numbers whose square starts and ends with exactly one 1.

Original entry on oeis.org

1, 11, 39, 41, 101, 111, 119, 121, 129, 131, 139, 141, 319, 321, 329, 331, 349, 351, 359, 361, 369, 371, 379, 381, 389, 391, 399, 401, 409, 411, 419, 421, 429, 431, 439, 441, 1001, 1009, 1011, 1019, 1021, 1029, 1031, 1039, 1041, 1099, 1101, 1109, 1111, 1119, 1121, 1129, 1131, 1139

Offset: 1

Bernard Schott, Oct 21 2021

When a square ends with 1, this square ends with exactly one 1.

Sequences A000533 and A253213 show that there are an infinity of terms. The square of their terms, for n >= 3, starts and ends with exactly one 1. Also, the numbers 119, 1119, 11119, ..., ((10^k + 71) / 9)^2, (k >= 3) are terms. The squares ((10^k + 71) / 9)^2, have the last digit 1 and because 12*10^(2*k - 3) < ((10^k + 71) / 9)^2 <13*10^(2*k - 3), for k >= 3, the squares ((10^k + 71) / 9)^2, k >= 4, start with 12. - Marius A. Burtea, Oct 21 2021

			39 is a term since 39^2 = 1521.
109 is not a term since 109^2 = 11881.
119 is a term since 119^2 = 14161.

Cf. A045855, A090771, A253213, A273372 (squares ending with 1), A017281, A017377.
Cf. A000533, A253213 for n >= 2 (subsequences).
Subsequence of A305719.

[1] cat [n:n in [2..1200]|Intseq(n*n)[1] eq 1 and Intseq(n*n)[#Intseq(n*n)] eq 1 and Intseq(n*n)[-1+#Intseq(n*n)] ne 1]; // Marius A. Burtea, Oct 21 2021

Join[{1}, Select[Range[11, 1200], (d = IntegerDigits[#^2])[[1]] == d[[-1]] == 1 && d[[2]] != 1 &]] (* Amiram Eldar, Oct 21 2021 *)

isok(k) = my(d=digits(sqr(k))); (d[1]==1) && (d[#d]==1) && if (#d>2, (d[2]!=1) && (d[#d-1]!=1), 1); \\ Michel Marcus, Oct 21 2021

from itertools import count, takewhile
def ok(n):
  s = str(n*n); return len(s.rstrip("1")) == len(s.lstrip("1")) == len(s)-1
def aupto(N):
  r = takewhile(lambda x: x<=N, (10*i+d for i in count(0) for d in [1, 9]))
  return [k for k in r if ok(k)]
print(aupto(1140)) # Michael S. Branicky, Oct 21 2021

A272673 Take list of squares that start with 1 (A045784) and omit the leading 1 and any leading zeros from what is left; if the number was a power of 10, replace it with 0.

Original entry on oeis.org

0, 6, 0, 21, 44, 69, 96, 24, 89, 156, 225, 296, 369, 444, 521, 600, 681, 764, 849, 936, 0, 201, 404, 609, 816, 1025, 1236, 1449, 1664, 1881, 2100, 2321, 2544, 2769, 2996, 3225, 3456, 3689, 3924, 4161, 4400, 4641, 4884, 5129, 5376, 5625, 5876

Offset: 1

Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 21 2016

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A045784, A045855.

A272673_list = [0] + [int(str(m**2)[1:]) if sum(int(d) for d in str(m**2)[1:]) != 1 else 0 for m in range(4,10**3) if str(m**2)[0] == '1'] # Chai Wah Wu, May 21 2016

cp's OEIS Frontend

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

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A045784 Squares with initial digit '1'.

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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

A348487 Positive numbers whose square starts and ends with exactly one 1.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

Python

A272673 Take list of squares that start with 1 (A045784) and omit the leading 1 and any leading zeros from what is left; if the number was a power of 10, replace it with 0.

Views

Author

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Python