A227070 - OEIS

%I #11 Jul 30 2013 14:56:12

%S 1,2,3,5,6,9,11,17,21,26,33,41,51,65,81,101,126,129,161,201,251,257,

%T 321,401,501,513,626,641,801,1001,1025,1251,1281,1601,2001,2049,2501,

%U 2561,3126,3201,4001,4097,5001,5121,6251,6401,8001,8193,10001

%N Powers n such that the set s(n) = {k > 0 such that k^n ends with k} does not occur for smaller n.

%C These numbers might be called automorphic powers because the sets s(n) are called automorphic numbers. It appears that all numbers of the form 1 + 5^i are here. In fact, these appear to produce the only even numbers here. The set s(4) equals s(2). The set s(7) equals s(3). The set s(9) does not differ from s(5) until k = 10443. The set s(17) does not differ from s(9) until k = 108307. The sequence also has 126, 201, 251, 501, and 626, but there may be missing numbers.

%C Entries a(17)-a(49) have been tentatively obtained by comparing the terms < 10^30 in the sets s(n), for 2 <= n <= 10001. - _Giovanni Resta_, Jul 30 2013

%F Conjecture: a(n+1) = A003592(n) + 1. - _Eric M. Schmidt_, Jul 30 2013

%t ts = {}; t = {}; Do[s = Select[Range[11000000], PowerMod[#, n, 10^IntegerLength[#]] == # &]; If[! MemberQ[ts, s], Print[n]; AppendTo[ts, s]; AppendTo[t, n]], {n, 2, 101}]; t = Join[{1}, t]

%Y Cf. A003226 (n=2), A033819 (n=3), A068407 (n=5), A068408 (n=6).

%Y Cf. A072496 (n=11), A072495 (n=21), A076650 (n=26).

%Y Cf. A227071.

%K nonn,hard,more,base

%O 1,2

%A _T. D. Noe_, Jul 29 2013

%E a(17)-a(49) from _Giovanni Resta_, Jul 30 2013