A161355 -id:A161355 - cp's OEIS frontend

A238712 Numbers in which squares may end (in base 10).

0, 1, 4, 5, 6, 9, 16, 21, 24, 25, 29, 36, 41, 44, 49, 56, 61, 64, 69, 76, 81, 84, 89, 96, 100, 104, 116, 121, 124, 129, 136, 144, 156, 161, 164, 169, 176, 184, 196, 201, 204, 209, 216, 224, 225, 236, 241, 244, 249, 256, 264, 276, 281, 284, 289, 296, 304

Offset: 1

M. F. Hasler, Mar 03 2014

The union of "squares mod 10" (= the first 6 terms) and "squares mod 100" (A010461) and "squares mod 1000" (A122986) etc.

The number of terms < 10^k beginning with k=0: 1, 6, 24, 165, 1101, 9306, 79620, 753462, 7198791, 70919559, ... - Robert G. Wilson v, Sep 04 2014

			6 is in the sequence because 4^2 = 16 ends in the digit 6.
7 is not in the sequence because no square can end with the digit 7.

Derek Orr, Table of n, a(n) for n = 1..9306

Cf. A161355, A246422, A246448 (Complement).

mx = 3; t = Union@ Table[ Mod[n^2, 10^mx], {n, 10^mx/2}]; t = Union@ Flatten@ Table[ Mod[t, 10^m], {m, mx}] (* Robert G. Wilson v, Sep 04 2014 *)

a=[];for(m=1,3,a=setunion(a,Set(vector(10^m,n,n^2)%10^m)));a

If n is present so is n^2. - Robert G. Wilson v, Sep 04 2014

A230604 Smallest number whose square has more than n digits and begins and ends with the same n digits.

Original entry on oeis.org

11, 173, 264, 16262, 193744, 238165, 38981039, 112791955, 1580178016, 1052631579, 30762132977, 15020242915, 14451789007487, 10909090909091, 1242844268897055, 1001889106154509, 4024018444782046, 10018891061545090, 11678332116788271168, 102040816316530612245, 139009056141395353279, 1128182832632197435939

Offset: 1

Arkadiusz Wesolowski, Feb 28 2014

Differs from A161355 in that the present sequence allows an "overlap" of the digits, while A161355 requires a(n)^2 to have at least 2n digits. - M. F. Hasler, Mar 03 2014

			a(3)=264 since 264^2 = 69696 is the smallest square that starts and ends with the same 3 digits.

Max Alekseyev, Table of n, a(n) for n = 1..32

Cf. A116501, A161355.

a[n_] := Block[{digits = {}},
  For[i = Ceiling[Sqrt[10^n]], True, i++,
   If[i^2 >= 10^n, digits = IntegerDigits[i^2];
    If[Take[digits, n] == Take[digits, -n], Return[i]]]]];
a2[#] & /@ Range[1, 6] (* Julien Kluge, Feb 02 2016 *)

for(n=1, 8, k=floor(sqrt(10^n)); until(bn==ed, k++; sr=Str(k^2); vc=Vec(sr); ln=#sr; bn=vc[1..n]; ed=vc[ln-n+1..ln]); print1(k, ", "));

a(9)-a(10) from Julien Kluge, Feb 13 2016
a(11)-a(12) from Julien Kluge, Mar 04 2016
a(13) from Giovanni Resta, Apr 18 2016
Terms a(14) onward from Max Alekseyev, Oct 11 2024

A202780 Smallest number whose cube begins and ends with the same n digits, and with any other digit(s) in between.

Original entry on oeis.org

7, 108, 335, 6667, 104636, 333335, 4504625, 70585736, 333333335, 1818181815, 33531510511, 333333333335, 198817904169

Offset: 1

Arkadiusz Wesolowski, Jan 06 2012

			335^3 = 37595375 starts and ends with the same 3 digits.

Cf. A161355.

lst = {}; Do[n = Ceiling[(10^(2*k))^(1/3)]; While[True, d = IntegerDigits[n^3]; If[Take[d, k] == Take[d, -k], Break[]]; n++]; AppendTo[lst, n], {k, 6}]; lst

a(9)-a(10) from Donovan Johnson, Jan 12 2012

a(11)-a(13) from Giovanni Resta, Apr 18 2016

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A238712 Numbers in which squares may end (in base 10).

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A230604 Smallest number whose square has more than n digits and begins and ends with the same n digits.

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Mathematica

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A202780 Smallest number whose cube begins and ends with the same n digits, and with any other digit(s) in between.

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Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Extensions