A255577 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.

2, 4, 5, 6, 8, 12, 16, 24, 25, 28, 32, 36, 44, 48, 52, 56, 64, 68, 72, 75, 76, 84, 88, 92, 96, 104, 112, 125, 128, 136, 144, 152, 168, 176, 184, 192, 208, 216, 224, 232, 248, 256, 264, 272, 288, 296, 304, 312, 328

Offset

1,1

Comments

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

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.

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

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

From Robert Israel, May 14 2015: (Start)

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.

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.

(End)

Links

Seyed Ali Tavakoli-Nabavi, Table of n, a(n) for n = 1..100000

Seyed Ali Tavakoli-Nabavi, Scatter plots: (n, a(n)) and (n, a(n) - a(n-1))

Seyed Ali Tavakoli-Nabavi, Tcl program to generate the b-file

Formula

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

Example

For n = 2, we have n^5 = 2^5 = 32, whose last digit is 2 = n, so 2 is in the sequence.
For n = 3, we have n^5 = 3^5 = 243, so 3 is in the sequence.
For n = 4, we have n^3 = 4^3 = 64, so 4 is in the sequence.
...
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.

Maple

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)):
sort(convert({seq(F(d), d=1..4)}, list)); # Robert Israel, May 14 2015

Prog

See a255577.tcl in LINKS

Crossrefs

Cf. A065502, A072495.

Sequence in context: A081354 A119792 A085834 * A245319 A037081 A270877

Adjacent sequences: A255574 A255575 A255576 * A255578 A255579 A255580

Keyword

nonn,base

Author

Seyed Ali Tavakoli-Nabavi, May 05 2015

Status

approved