The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A343034 Positive numbers m such that m^2 with last digit z deleted is still a perfect square k^2, and z divides m-k. 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 (list; graph; refs; listen; history; text; internal format)
 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 m-k = 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 = m-k: (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 m-k. 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 m-k. 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 m-k. First few solutions (m, k) are (13, 4), (487, 154), (18493, 5848), ... with m = A228209(3q). LINKS Table of n, a(n) for n=1..24. Diophante, A1872, Carrément magiques (in French). Index entries for linear recurrences with constant coefficients, signature (0,38,0,-1). FORMULA a(2n+1) = A078986(n) for n >= 0. a(2n) = A228209(3n) for n >= 1. 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 = 13-4 divides 13-4. For m = 19, 19^2 = 361, 6^2 = 36, 19^2 - 10*6^2 = 1 and 1 divides 19-6 = 13. For m = 487, 487^2 = 237169, 154^2 = 23716, 487^2 - 10*154^2 = 9 and 9 divides 487-154 = 333 = 9*37. MATHEMATICA LinearRecurrence[{0, 38, 0, -1}, {1, 13, 19, 487}, 24] (* Amiram Eldar, Apr 03 2021 *) CROSSREFS Subsequence of A031149. A078986 is a subsequence. Cf. A078987, A228209. Sequence in context: A147393 A088187 A146047 * A145944 A355964 A330425 Adjacent sequences: A343031 A343032 A343033 * A343035 A343036 A343037 KEYWORD nonn,easy,base AUTHOR Bernard Schott, Apr 03 2021 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 5 19:06 EDT 2023. Contains 363138 sequences. (Running on oeis4.)