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

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

Offset: 1

Author

Erich Friedman, Jun 23 2001

Keywords

• base
• nonn

Comments

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

Examples

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

Links

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

Crossrefs

Subsequence of A117755.

Cf. A002275, A061844.

Programs

Mathematica

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 *)

PARI

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

Extensions

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