A116501 -id:A116501 - cp's OEIS frontend

A305719 Numbers whose squares have the same first and last digits.

1, 2, 3, 11, 22, 26, 39, 41, 68, 75, 97, 101, 109, 111, 119, 121, 129, 131, 139, 141, 202, 208, 212, 218, 222, 225, 235, 246, 254, 256, 264, 303, 307, 313, 319, 321, 329, 331, 339, 341, 349, 351, 359, 361, 369, 371, 379, 381, 389, 391, 399, 401, 409, 411, 419, 421, 429, 431, 439, 441, 638

Offset: 1

Neville Holmes, Jun 08 2018

			For k = 11, k^2 = 121;
for k = 26, k^2 = 676.

Cf. A227858, A116499, A116501.

Cf. A123912, A186438.

Select[Range[638], (d = IntegerDigits[#^2]; d[[1]] == d[[-1]]) &] (* Giovanni Resta, Jun 25 2018 *)

for(n=1, 10^3, my(d=digits(n^2)); if( d[1]==d[#d], print1(n,", "))); \\ Joerg Arndt, Jun 10 2018

def ok(n): s = str(n*n); return s[0] == s[-1]
print(list(filter(ok, range(1, 639)))) # Michael S. Branicky, Jul 16 2021

A116499 Numbers k such that the base-10 representation of k^2 has the structure dmmd, where d is a single digit and m is a string of digits.

Original entry on oeis.org

63565788, 777035126, 785900449264170834, 824735400720481514, 41969556121016094021, 7841042904081599033746826, 775474811976872476373043200886, 403578381989727694739321538082071713672671, 820657465670154825723396137907157085701046

Offset: 1

Giovanni Resta, Feb 18 2006

			785900449264170834^2 = 6[17639516153625555][17639516153625555]6.

Max Alekseyev, Table of n, a(n) for n = 1..100

Cf. A116500, A116501, A305719.

A116502 Squares with structure dmdmd, where d is a single digit and m a string of digits.

Original entry on oeis.org

69696, 56722567225, 95540955409, 1108033241108033241, 1220096161220096161, 4312080784312080784, 4432132964432132964, 4880384644880384644, 9024307889024307889, 9972299169972299169, 946308825294630882529, 11542927396115429273961, 46171709584461717095844

Offset: 1

Giovanni Resta, Feb 18 2006

			309097^2 = a(3) = 9[5540]9[5540]9.

Giovanni Resta, Table of n, a(n) for n = 1..300
Erich Friedman, Puzzles of the Week [_N. J. A. Sloane_, Mar 12 2011]

Cf. A116500, A116501, A069919.

w={}; Do[s = Reduce[(1 + 10^(1+e) + 100^(1+e)) d + 10 (1+10^(1+e)) x == y^2 && 0 <= x < 10^e && y>0, {x,y}, Integers]; If[s =!= False, w = Union[w, y^2 /. List@ ToRules@ s]], {e, 12}, {d, 9}]; w (* Giovanni Resta, Aug 01 2018 *)

a(n) = A116501(n)^2.

a(11)-a(13) from Giovanni Resta, Aug 01 2018

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

A216233 Numbers whose squares have 2R-1 digits, such that the number represented by leftmost R digits and number represented by rightmost R digits divide each other evenly.

Original entry on oeis.org

245, 249, 251, 255, 264, 1245, 1249, 2490, 2498, 2502, 2510, 10984, 12490, 12498, 15449, 18735, 18751, 18868, 22714, 24980, 24996, 27907, 28302, 31225, 31249, 31579, 101852, 124996, 139535, 152174, 187494, 187510, 218751, 238165, 249992, 279070, 281249

Offset: 1

Thomas S. Pedigo, Mar 14 2013

Trivial solutions where the rightmost R digits are all zeros have been omitted. The first indices k for which the rightmost R digits of a(k)^2 do not contain leading zeros are 5, 12, 15, 19, 26, 27, 30, 34, 39, 52, 53, 62, 67, 80.

			The square of 22714 is 515925796, and 51592 = 2*25796.

Giovanni Resta, Table of n, a(n) for n = 1..165

Cf. A115528, A116501, A145848.

cnt = 0; Do[p = 10^Floor[nd/2]; Do[x = Floor[n*n/p]; y = Mod[n*n, 10*p]; If[y>0 && Mod[x,y]*Mod[y,x] == 0, Print[++cnt, " ", n, " ", n*n]], {n, p, Floor[10^(nd/2)]}], {nd,3,11,2}] (* Giovanni Resta, Mar 15 2013 *)

Missing a(25) and a(27)-a(37) from Giovanni Resta, Mar 15 2013

Comment corrected by Giovanni Resta, Mar 15 2013

cp's OEIS Frontend

A305719 Numbers whose squares have the same first and last digits.

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A116499 Numbers k such that the base-10 representation of k^2 has the structure dmmd, where d is a single digit and m is a string of digits.

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A116502 Squares with structure dmdmd, where d is a single digit and m a string of digits.

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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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Extensions

A216233 Numbers whose squares have 2R-1 digits, such that the number represented by leftmost R digits and number represented by rightmost R digits divide each other evenly.

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