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

0, 0, 2, 3, 6, 57, 141, 390, 8790, 82128, 11963, 793212, 835571, 5054322, 1661682, 75880433, 45322418, 619541169, 319375992, 6745157241, 3317093849, 66891312600, 843114912509, 9837094694375, 16065496578813, 35901922360062, 67557477392256, 547721051611007

Offset: 1

Author

Christopher Hohl, Mar 24 2019

Keywords

• nonn
• base

Comments

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.

In such cases, it also appears that a(n)*a(n+1) = k^2 - k.

Examples

			For n=4, A003226(4)=6, (A003226(4))^2=36. So a(4)=3.
For n=13, A003226(13)=2890625, (A003226(13))^2=8355712890625. So a(13)=835571.
		

Crossrefs

Cf. A003226, A055642.

Programs

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
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

Formula

a(n) = A003226(n)*(A003226(n) - 1) / 10^A055642(A003226(n)).