

A343034


Positive numbers m such that m^2 with last digit z deleted is still a perfect square k^2, and z divides mk.


0



1, 13, 19, 487, 721, 18493, 27379, 702247, 1039681, 26666893, 39480499, 1012639687, 1499219281, 38453641213, 56930852179, 1460225726407, 2161873163521, 55450123962253, 82094249361619, 2105644484839207, 3117419602578001, 79959040299927613, 118379850648602419, 3036337886912410087
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OFFSET

1,2


COMMENTS

This sequence is the answer to the problem A1872 proposed on French mathematical site Diophante (see link).
Equivalent to the two Diophantine equations: m^2 = 10*k^2 + z and mk = q*z for some q >= 1.
There exist solutions iff z = 1 or z = 9.
When (m, k, z) is a solution, then (19m+60k, 6m+19k, z) is another solution.
There is only one solution such that z = mk: (13, 4, 9), see 1st example.
There exist two distinct families of solutions corresponding to z = 1 and z = 9, odd indices correspond to z = 1 and even indices to z = 9.
> For z = 1, all solutions m of Pell equation m^2  10*k^2 = 1 are terms because z = 1 divides every mk.
First few solutions (m, k) are (1, 0), (19, 6), (721, 228), (27379, 86568), ... with m = A078986(q) and corresponding k = 6*A078987(q).
> For z = 9, solutions m must satisfy m^2  10*k^2 = 9 with 9 divides mk. Among the 3 fundamental solutions (3, 0), (7, 2), (13, 4) of Pell equation m^2 10*k^2 = 9, only (13, 4) gives solutions where 9 divides mk.
First few solutions (m, k) are (13, 4), (487, 154), (18493, 5848), ... with m = A228209(3q).


LINKS



FORMULA

a(n+4) = 38*a(n+2)  a(n), a(1) = 1, a(2) = 13, a(3)= 19, a(4) = 487.
G.f.: x*(1 + 13*x  19*x^2  7*x^3)/(1  38*x^2 + x^4).  Stefano Spezia, Apr 03 2021


EXAMPLE

For m = 13, 13^2 = 169, 4^2 = 16, 13^2  10*4^2 = 9 and 9 = 134 divides 134.
For m = 19, 19^2 = 361, 6^2 = 36, 19^2  10*6^2 = 1 and 1 divides 196 = 13.
For m = 487, 487^2 = 237169, 154^2 = 23716, 487^2  10*154^2 = 9 and 9 divides 487154 = 333 = 9*37.


MATHEMATICA

LinearRecurrence[{0, 38, 0, 1}, {1, 13, 19, 487}, 24] (* Amiram Eldar, Apr 03 2021 *)


CROSSREFS



KEYWORD

nonn,easy,base


AUTHOR



STATUS

approved



