cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

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

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Magma
    [n: n in [1..10000 ] | IsSquare(Seqint(Intseq(n) cat Intseq(1)))]; // Marius A. Burtea, Mar 21 2019
  • Maple
    t1:=[];
for k from 1 to 20000 do
if issqr(k+10^length(k)) then t1:=[op(t1),k]; fi;
od;
t1;
  • Mathematica
    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 *)
  • PARI
    isok(n) = issquare(eval(concat(1, Str(n)))); \\ Michel Marcus, Mar 21 2019
  • Python
    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

Extensions

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

Views

Author

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

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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.

Programs

  • Mathematica
    << 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 *)

A272684 Terms of A272671 which are not a power of 100 times an earlier term of A272671.

Original entry on oeis.org

6, 21, 44, 69, 96, 156, 225, 296, 369, 444, 521, 681, 764, 849, 936, 1025, 1236, 1449, 1664, 1881, 2321, 2544, 2769, 2996, 3225, 3456, 3689, 3924, 4161, 4641, 4884, 5129, 5376, 5625, 5876, 6129, 6384, 6641, 7161, 7424, 7689, 7956, 8225, 8496, 8769, 9044, 9321, 9881
Offset: 1

Views

Author

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

Keywords

Comments

These are the only terms in A272671 that are candidates for terms of A265432.

Crossrefs

Cf. A272671, A265432, A272685.

A273369 a(n) is the smallest m such that A265432(m) = A272671(n), or -1 if no such m exists.

Original entry on oeis.org

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

Views

Author

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

Keywords

Comments

Every entry in A265432 appears in A272671.
a(n) = -1 whenever A272671(n) ends in 0, because every such entry ends in 00 and anytime concat(1,k,00) and concat(n,k,00) are both perfect squares, concat(1,k) and concat(n,k) are also both perfect squares.
Does every term in A272671 that does not end in 0 appear in A265432?
See A272685 for a version that takes into account the fact that terms in A272671 ending in 0 cannot appear in A265432.

Examples

			A272671(6) = 156. A265432(217) = 156, but A265432(m) does not equal 156 for any m < 217. So a(6) = 217.

Links

Crossrefs

Cf. A265432, A272671.
See A272685 for another version.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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