A227070 Powers n such that the set s(n) = {k > 0 such that k^n ends with k} does not occur for smaller n.

1, 2, 3, 5, 6, 9, 11, 17, 21, 26, 33, 41, 51, 65, 81, 101, 126, 129, 161, 201, 251, 257, 321, 401, 501, 513, 626, 641, 801, 1001, 1025, 1251, 1281, 1601, 2001, 2049, 2501, 2561, 3126, 3201, 4001, 4097, 5001, 5121, 6251, 6401, 8001, 8193, 10001

Offset

1,2

Comments

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.

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

Links

Table of n, a(n) for n=1..49.

Formula

Conjecture: a(n+1) = A003592(n) + 1. - Eric M. Schmidt, Jul 30 2013

Mathematica

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]

Crossrefs

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

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

Cf. A227071.

Sequence in context: A363066 A060714 A241819 * A032718 A366143 A086191

Adjacent sequences: A227067 A227068 A227069 * A227071 A227072 A227073

Keyword

nonn,hard,more,base

Author

T. D. Noe, Jul 29 2013

Extensions

a(17)-a(49) from Giovanni Resta, Jul 30 2013

Status

approved