%I #63 Sep 02 2024 19:33:35

%S 2,4,5,6,8,12,16,24,25,28,32,36,44,48,52,56,64,68,72,75,76,84,88,92,

%T 96,104,112,125,128,136,144,152,168,176,184,192,208,216,224,232,248,

%U 256,264,272,288,296,304,312,328

%N Numbers n not coprime to 10 such that there exists an integer k > 1 where n^k contains n as its last digits in base 10.

%C Intersection of A065502 (numbers not coprime with 10) and A072495 (k-morphic numbers).

%C Also defined as all n not coprime with 10 where there exists k > 1 such that n^k mod 10^floor(log_10(n)) = n.

%C For n > 1, a(n) <= a(n-1) + 2^(ceiling(log_10(a(n))) + 1) (conjectured).

%C For a(n) >= 10^k where k >= 1, there exists a(m) = a(n) mod 10^j where m < n and j < k.

%C From _Robert Israel_, May 14 2015: (Start)

%C n with d digits is in the sequence if and only if n is either divisible by 2^d but not by 5, or divisible by 5^d but not by 2.

%C For d >= 2 the number of terms with d digits is 4*5^(d-1) + 2^(d-1) - 4*floor(5^d/50) - floor(2^d/20) - x(d) where x(d) = 3 if d == 2 or 3 mod 4, 2 otherwise.

%C (End)

%H Seyed Ali Tavakoli-Nabavi, <a href="/A255577/b255577.txt">Table of n, a(n) for n = 1..100000</a>

%H Seyed Ali Tavakoli-Nabavi, <a href="/A255577/a255577.png">Scatter plots: (n, a(n)) and (n, a(n) - a(n-1))</a>

%H Seyed Ali Tavakoli-Nabavi, <a href="/A255577/a255577.tcl.txt">Tcl program to generate the b-file</a>

%F For n > 6, a(n) > 2n since no term is divisible by 10 (but all are divisible by either 2 or 5). - _Charles R Greathouse IV_, May 13 2015

%e For n = 2, we have n^5 = 2^5 = 32, whose last digit is 2 = n, so 2 is in the sequence.

%e For n = 3, we have n^5 = 3^5 = 243, so 3 is in the sequence.

%e For n = 4, we have n^3 = 4^3 = 64, so 4 is in the sequence.

%e ...

%e As a counterexample, n = 41 is not in the sequence because it is coprime with 10, even though we have 41^6 = 4750104241, whose last 2 digits are 41.

%p F:= d -> (seq(seq(2^d*(5*j+i),i=1..4),j=0..5^(d-1)-1), seq(5^d*(2*j+1),j=0..2^(d-1)-1)):

%p sort(convert({seq(F(d),d=1..4)},list)); # _Robert Israel_, May 14 2015

%o (Tcl) See a255577.tcl in LINKS.

%Y Cf. A065502, A072495.

%K nonn,base

%O 1,1

%A _Seyed Ali Tavakoli-Nabavi_, May 05 2015