A061844 -id:A061844 - cp's OEIS frontend

A255850 Number of digits in the n-th term of A061844: Squares which remain squares if you decrease every digit by 1.

1, 2, 4, 5, 7, 11, 12, 18, 20, 20, 20, 21, 30, 30, 30, 30, 30, 30, 30, 30, 32, 32, 32, 40, 42, 42, 42, 42, 42, 42, 42, 42, 42, 54, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60

Offset: 1

M. F. Hasler, Mar 10 2015

Suggested by W. Appleby in a Number Theory discussion group, cf. link.

Lars Blomberg, Table of n, a(n) for n = 1..78
G. Campbell et al., A six digit problem, Number Theory group on LinkedIn.com, March 2015.

Cf. A061844, A055642, A255851.

a(n)=#Str(A061844(n)) \\ if A061844 is a function

A255850=apply(t->#Str(t),A061844) \\ if A061844 is a vector

a(n) = A055642(A061844(n)).

More terms calculated from A061844 by Lars Blomberg, Aug 21 2016

A255851 Number of n-digit squares which remain squares if every digit is decreased by 1 (cf. A061844).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 34, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0

Offset: 1

M. F. Hasler, Mar 10 2015

Counts the occurrences of n in A255850.

Lars Blomberg, Table of n, a(n) for n = 1..180

Cf. A255850, A061844, A055642.

PARI
```
a(n,A=A255850)=sum(i=1,#A,A[i]==n)
```

Corrected a(30) and more terms calculated from A061844 by Lars Blomberg, Aug 21 2016

A117755 Squares which remain squares when each digit is replaced by the next digit.

Original entry on oeis.org

0, 9, 25, 2025, 13225, 1974025, 4862025, 6943225, 60415182025, 207612366025, 916408817340025, 9960302475729225, 153668543313582025, 1978088677245614025, 13876266042653742025, 20761288044852366025, 47285734107144405625, 406066810454367265225

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), Apr 14 2006

Replace 1 with 2, 2 with 3, ..., 8 with 9 and 9 with 0.

			13225 is in the sequence because (1) it is a square and (2) if we transform it we get 24336 and this is also a square.

Max Alekseyev, Table of n, a(n) for n = 1..22

Cf. A048379, A061843, A061844, A127856.

Select[Range[0, 500000]^2, IntegerQ[Sqrt[FromDigits[(1 + IntegerDigits[ # ]) /. 10-> 0]]] &] (* Harvey P. Dale, Jan 21 2007 *)

More terms from Harvey P. Dale, Jan 21 2007
3 more terms from Donovan Johnson, Apr 03 2008
a(14)-a(21) from Max Alekseyev, Oct 22 2008
a(1) = 0 prepended by Max Alekseyev, Jul 26 2023

A273234 Squares that remain squares if you decrease them by 8 times a repunit with the same number of digits.

Original entry on oeis.org

9, 889249, 896809, 908209, 902942754289, 924745719769, 946618081249, 987107822089, 910909843526089, 9810767198166489, 888909576913320169, 889214944824055249, 889286612895723249, 889972999762742809, 890923059538260849, 896642235371330809, 896979367708462809

Offset: 1

Paolo P. Lava, May 18 2016

Any number ends in 9.

			9 - 8*1 = 1 = 1^2;
889249 - 8*111111 = 361 = 19^2;
896809 - 8*111111 = 7921 = 89^2.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A002275, A061844, A273299-A273233.

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(10^9,8);

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_] := 8 (10^n-1)/9; Flatten[sol /@ r /@ Range[12]] (* Giovanni Resta, May 18 2016 *)

a(11)-a(17) from Giovanni Resta, May 18 2016

A061843 Squares which remain squares if you increment every digit by 1.

Original entry on oeis.org

0, 25, 2025, 13225, 4862025, 60415182025, 207612366025, 153668543313582025, 13876266042653742025, 20761288044852366025, 47285734107144405625, 406066810454367265225, 141704161680410868660551655625

Offset: 1

Erich Friedman, Jun 23 2001

Incrementing each digit means b^2-a^2 = R_n, the n-digit repunit (10^n-1)/9; so solutions must be of the form a = (u-v)/2, b = (u+v)/2, where u * v = R_n. It remains to check that this is in the right range and a has no 9's. - Franklin T. Adams-Watters, May 25 2006

			13225 = 115^2 and 24336 = 156^2.

Max Alekseyev, Table of n, a(n) for n = 1..78 (contains all terms below 10^262)

Subsequence of A117755.

Cf. A002275, A061844.

Select[Range[0,500000]^2,With[{lst=IntegerDigits[#]+1},Max[lst]<10&&IntegerQ[Sqrt[FromDigits[lst]]]&]] (* The program generates the first 7 terms of the sequence. *)  (* Harvey P. Dale, Jan 26 2025 *)

hasdigit(n, d, b=10) = local(r); r=0;while(r==0&&n>=1,if(n%b==d,r=1,n\=b));r
/* Generates all positive n-digit solutions (in reverse order) */
A061843s(n) = local(f, nf, v, i, ru, lb, ub, x); lb=10^(n-1);ub=10^n-1;ru=ub\9;f=divisors(ru);v=[];nf=matsize(f)[2];for(i=1,nf\2,x=( (f[nf+1-i]-f[i])\2)^2;if(x>=lb&&x<=ub&&!hasdigit(x,9),v=concat(v,[x])));v \\ Franklin T. Adams-Watters, May 25 2006

More terms from Franklin T. Adams-Watters, May 25 2006

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

Paolo P. Lava, May 18 2016

Apart from the initial term, any number ends in 0 or 6.

			1 - 1 = 0 = 0^2;
36 - 11 = 25 = 5^2;
400 - 111 = 289 = 17^2;

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A002275, A061844, A273230-A273234.

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

Paolo P. Lava, May 18 2016

Apart from the initial term, any number ends in 9.

			4 - 3*1 = 1 = 1^2;
49 - 3*11 = 16 = 4^2;
529 - 3*111 = 196 = 14^2.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A002275, A061844, A273229, A273231-A273234.

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

Paolo P. Lava, May 18 2016

Every term ends in 0 or 4.

			4 - 4*1 = 0 = 0^2;
97344 - 4*11111 = 52900 = 230^2;
462400 - 4*111111 = 17956 = 134^2.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A002275, A061844, A273299, A273230, A273232-A273234.

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 *)

a(16)-a(19) from 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

Paolo P. Lava, May 18 2016

Apart from the initial term, any number ends in 4 or 6.

			9 - 5*1 = 4 = 2^2;
64 - 5*11 = 9 = 3^2;
676 - 5*111 = 121 = 11^2.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A002275, A061844, A273299-A273231, A273233, A273234.

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

Paolo P. Lava, May 18 2016

Any number ends in 1.

			81 - 7*11 = 4 = 2^2;
841 - 7*111 = 64 = 8^2;
7921 - 7*1111 = 144 = 12^2.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A002275, A061844, A273299-A273232, A273234.

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 *)

cp's OEIS Frontend

A255850 Number of digits in the n-th term of A061844: Squares which remain squares if you decrease every digit by 1.

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A255851 Number of n-digit squares which remain squares if every digit is decreased by 1 (cf. A061844).

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A117755 Squares which remain squares when each digit is replaced by the next digit.

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A273234 Squares that remain squares if you decrease them by 8 times a repunit with the same number of digits.

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A061843 Squares which remain squares if you increment every digit by 1.

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A273229 Squares that remain squares if you decrease them by a repunit with the same number of digits.

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Maple

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A273230 Squares that remain squares if you decrease them by 3 times a repunit with the same number of digits.

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Maple

Mathematica