A273229
Squares that remain squares if you decrease them by a repunit with the same number of digits.
Original entry on oeis.org
1, 36, 400, 3136, 24336, 115600, 118336, 126736, 211600, 309136, 430336, 577600, 5973136, 19713600, 30869136, 53582400, 3086469136, 4310710336, 71526293136, 111155560000, 112104432400, 113531259136, 137756776336, 206170483600, 245996160400, 262303768336, 308642469136
Offset: 1
1 - 1 = 0 = 0^2;
36 - 11 = 25 = 5^2;
400 - 111 = 289 = 17^2;
-
P:=proc(q,h) local n; for n from 1 to q do
if type(sqrt(n^2-h*(10^(ilog10(n^2)+1)-1)/9),integer) then print(n^2);
fi; od; end: P(10^9,1);
-
sol[k_] := Block[{x, e = IntegerLength@k, d = Divisors@k}, Union[#+k/# & /@ Select[ Take[d, Ceiling[ Length@d/2]], EvenQ[ x= #+k/#] && IntegerLength[ x^2/4] == e &]]^2/4]; r[n_] := (10^n-1)/9; Flatten[sol /@ r /@ Range[12]] (* Giovanni Resta, May 18 2016 *)
A273230
Squares that remain squares if you decrease them by 3 times a repunit with the same number of digits.
Original entry on oeis.org
4, 49, 529, 4489, 38809, 344569, 363609, 375769, 444889, 558009, 597529, 700569, 7198489, 35366809, 44448889, 65983129, 4444488889, 5587114009, 83574762649, 335330171929, 359763638809, 390241344249, 403831017529, 407200963129, 435775577689, 444444888889, 453557800089
Offset: 1
4 - 3*1 = 1 = 1^2;
49 - 3*11 = 16 = 4^2;
529 - 3*111 = 196 = 14^2.
-
P:=proc(q,h) local n; for n from 1 to q do
if type(sqrt(n^2-h*(10^(ilog10(n^2)+1)-1)/9),integer) then print(n^2);
fi; od; end: P(10^9,3);
-
sol[k_] := Block[{x, e = IntegerLength@k, d = Divisors@ k}, Union[ #+k/# & /@ Select[ Take[d, Ceiling[ Length@d/2]], EvenQ[x = #+k/#] && IntegerLength[ x^2/4] == e &]]^2/4]; r[n_] := 3 (10^n-1)/9; Flatten[sol /@ r /@ Range[12]] (* Giovanni Resta, May 18 2016 *)
A273231
Squares that remain squares if you decrease them by 4 times a repunit with the same number of digits.
Original entry on oeis.org
4, 97344, 462400, 473344, 506944, 846400, 78854400, 444622240000, 448417729600, 454125036544, 551027105344, 824681934400, 983984641600, 460651783840000, 6703941381760000, 444446222224000000, 459134832243732544, 462218702574222400, 462583182938702400
Offset: 1
4 - 4*1 = 0 = 0^2;
97344 - 4*11111 = 52900 = 230^2;
462400 - 4*111111 = 17956 = 134^2.
-
P:=proc(q,h) local n; for n from 1 to q do
if type(sqrt(n^2-h*(10^(ilog10(n^2)+1)-1)/9),integer) then print(n^2);
fi; od; end: P(10^9,4);
-
sol[k_] := Block[{x, e = IntegerLength@k, d = Divisors@ k}, Union[ #+k/# & /@ Select[ Take[d, Ceiling[ Length@d/2]], EvenQ[x = #+k/#] && IntegerLength[ x^2/4] == e &]]^2/4]; r[n_] := 4 (10^n-1)/9; Flatten[sol /@ r /@ Range[12]] (* Giovanni Resta, May 18 2016 *)
A273232
Squares that remain squares if you decrease them by 5 times a repunit with the same number of digits.
Original entry on oeis.org
9, 64, 676, 6084, 56644, 556516, 605284, 669124, 702244, 743044, 784996, 835396, 8538084, 55562116, 60497284, 79673476, 6049417284, 7028810244, 96560590564, 555838838116, 567620600836, 575774404804, 604938617284, 612115334884, 619365852004, 643617898564, 817422124996
Offset: 1
9 - 5*1 = 4 = 2^2;
64 - 5*11 = 9 = 3^2;
676 - 5*111 = 121 = 11^2.
-
P:=proc(q,h) local n; for n from 1 to q do
if type(sqrt(n^2-h*(10^(ilog10(n^2)+1)-1)/9),integer) then print(n^2);
fi; od; end: P(10^9,5);
-
sol[k_] := Block[{x, e = IntegerLength@k, d = Divisors@ k}, Union[ #+k/# & /@ Select[ Take[d, Ceiling[ Length@d/2]], EvenQ[x = #+k/#] && IntegerLength[ x^2/4] == e &]]^2/4]; r[n_] := 5 (10^n-1)/9; Flatten[sol /@ r /@ Range[12]] (* Giovanni Resta, May 18 2016 *)
A273233
Squares that remain squares if you decrease them by 7 times a repunit with the same number of digits.
Original entry on oeis.org
81, 841, 7921, 77841, 790321, 863041, 982081, 9991921, 79014321, 80299521, 94653441, 7901254321, 8635799041, 778133930161, 790123654321, 794396081521, 816057482881, 965485073281, 989863816561, 79012347654321, 86358529399041, 857789228465521, 7901234587654321, 8547733055510401
Offset: 1
81 - 7*11 = 4 = 2^2;
841 - 7*111 = 64 = 8^2;
7921 - 7*1111 = 144 = 12^2.
-
P:=proc(q,h) local n; for n from 1 to q do
if type(sqrt(n^2-h*(10^(ilog10(n^2)+1)-1)/9),integer) then print(n^2);
fi; od; end: P(10^9,7);
-
sol[k_] := Block[{x, e = IntegerLength@k, d = Divisors@ k}, Union[ #+k/# & /@ Select[ Take[d, Ceiling[ Length@d/2]], EvenQ[x = #+k/#] && IntegerLength[ x^2/4] == e &]]^2/4]; r[n_] := 7 (10^n-1)/9; Flatten[sol /@ r /@ Range[12]] (* Giovanni Resta, May 18 2016 *)
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