A350870 Numbers k = x.y such that x.y = (x+y)^2, when x and y have the same number of digits, "." means concatenation, and y may not begin with 0.

81, 2025, 3025, 494209, 24502500, 25502500, 52881984, 60481729, 6049417284, 6832014336, 101558217124, 108878221089, 123448227904, 127194229449, 152344237969, 213018248521, 217930248900, 249500250000, 250500250000, 284270248900, 289940248521, 371718237969, 413908229449, 420744227904

Offset

1,1

Comments

Problem proposed on French site Diophante (see link).

We have to solve Diophantine equation (x+y)^2 = x*10^m + y where m = length(x) = length(y).

Squares of a variant of Kaprekar numbers (A045913).

Number of solutions with 2*m digits for m >= 1: 1, 2, 1, 4, 2, 21, ...

Compare with A347541 where x*y divides x.y, when x and y have the same number of digits, "." means concatenation, and y may not begin with 0.

Links

Table of n, a(n) for n=1..24.

Diophante, A1945 - Concaténations en tous genres (in French).

Formula

a(n) = A045913(n+1)^2.

Example

81 = (8+1)^2, hence 81 is a term.
3025 = (30+25)^2, hence 3025 is another term.

Prog

(PARI) upto(n) = {i = 4; i2 = i^2; res = List(); while(i2 <= n, i++; i2 = i^2; if(#digits(i2) % 2 == 1, i = sqrtint(10^(#digits(i2))) + 1; i2 = i^2; ); if(is(i2), listput(res, i2) ); ); res }
is(n) = { my(d = digits(n), d1, d2, frd2); if(#d % 2 == 1, return(0)); d1 = vector(#d \ 2, i, d[i]); d2 = vector(#d \ 2, i, d[i + #d \ 2]); frd2 = fromdigits(d2); 10^(#d \ 2 - 1) <= frd2 && (fromdigits(d1) + frd2)^2 == n } \\ David A. Corneth, Jan 21 2022

Crossrefs

Equals A238237 \ A350918.

Subsequence of A102766.

Subsequences: A038544, A350869.

Cf. A045913, A347541.

Sequence in context: A205978 A037211 A238237 * A282861 A237376 A296577

Adjacent sequences: A350867 A350868 A350869 * A350871 A350872 A350873

Keyword

nonn,base

Author

Bernard Schott, Jan 20 2022

Status

approved