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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 27 15:14 EST 2020. Contains 331295 sequences. (Running on oeis4.)