cp's OEIS Frontend

A307104 a(n) is the number which, when concatenated with A003226(n), the n-th automorphic number, gives (A003226(n))^2.

%I A307104 #56 Dec 05 2024 18:58:46
%S A307104 0,0,2,3,6,57,141,390,8790,82128,11963,793212,835571,5054322,1661682,
%T A307104 75880433,45322418,619541169,319375992,6745157241,3317093849,
%U A307104 66891312600,843114912509,9837094694375,16065496578813,35901922360062,67557477392256,547721051611007
%N A307104 a(n) is the number which, when concatenated with A003226(n), the n-th automorphic number, gives (A003226(n))^2.
%C A307104 Let na and nb represent the indices of the preceding and next A003226(n)'s beginning with a 9, and where (na - nb) >= 3 (note that the first such 'zone' begins with an exception for which the index A003226(na) = 1). Then for na < n < nb and such that n == (na + 1) mod 2, it appears that A003226(n) - a(n) = A003226(n+1) - a(n+1) = k.
%C A307104 In such cases, it also appears that a(n)*a(n+1) = k^2 - k.
%F A307104 a(n) = A003226(n)*(A003226(n) - 1) / 10^A055642(A003226(n)).
%e A307104 For n=4, A003226(4)=6, (A003226(4))^2=36. So a(4)=3.
%e A307104 For n=13, A003226(13)=2890625, (A003226(13))^2=8355712890625. So a(13)=835571.
%o A307104 (PARI) auto(n) = {n<3 & return(n-1); my(i=10, j=10, b=5, c=6, a=b); for( k=4, n, while(b<=a, b=b^2%i*=10); while(c<=a, c=(2-c)*c%j*=10); a=min(b, c)); a; } \\ A003226
%o A307104 a(n) = {my(m = auto(n), dm = digits(m), dm2 = digits(m^2)); fromdigits(vector(#dm2 - #dm, k, dm2[k]));} \\ _Michel Marcus_, May 18 2019
%Y A307104 Cf. A003226, A055642.
%K A307104 nonn,base
%O A307104 1,3
%A A307104 _Christopher Hohl_, Mar 24 2019