A353441 - OEIS

%I #32 Jul 01 2023 16:54:19

%S 2,4,7,8,14,16,17,18,19,20,22,23,26,28,29,31,32,34,35,38,39,40,42,43,

%T 46,47,49,51,53,54,56,57,58,59,61,62,63,64,65,66,67,68,69,70,71,74,76,

%U 78,79,80,81,82,83,85,86,87,89,92,93,94,95,97,98,102,103,104,105,106,107,108,109

%N Integers m such that the decimal expansion of 1/m contains the digit 5.

%C If m is a term, 10*m is also a term, so terms with no trailing zeros are all primitive terms.

%H Robert Israel, <a href="/A353441/b353441.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/1#1overn">Index entries for sequences related to decimal expansion of 1/n</a>.

%e m = 7 is a term since 1/7 = 0.142857142857...

%e m = 22 is a term since 1/22 = 0.04545454545... (here, 5 is the largest digit).

%e m = 132 is a term since 1/693 = 0.00757575... (here, 5 is the smallest digit).

%p filter:= proc(n) local q;

%p   q:= NumberTheory:-RepeatingDecimal(1/n);

%p   member(5,RepeatingPart(q)) or member(5, NonRepeatingPart(q))

%p end proc:

%p select(filter, [$1..200]); # _Robert Israel_, Apr 25 2023

%t f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 125, MemberQ[f@#, 5] &]

%o (Python)

%o from itertools import count, islice

%o from sympy import multiplicity, n_order

%o def A353441_gen(startvalue=1): # generator of terms >= startvalue

%o     for a in count(max(startvalue,1)):

%o         m2, m5 = (~a&a-1).bit_length(), multiplicity(5,a)

%o         k, m = 10**max(m2,m5), 10**n_order(10,a//(1<<m2)//5**m5)-1

%o         if '5' in str(c:=k//a) or '5' in str(m*k//a-c*m):

%o             yield a

%o A353441_list = list(islice(A353441_gen(),20)) # _Chai Wah Wu_, May 01 2023

%Y A351471 (largest digit=5) and A352159 (smallest digit=5) are subsequences.

%Y Similar with digit k: A352154 (k=0), A353437 (k=1), A353438 (k=2), A353439 (k=3), A353440 (k=4), this sequence (k=5), A353442 (k=6), A353443 (k=7), A353444 (k=8), A333237 (k=9).

%Y Complement of A362579.

%K nonn,base

%O 1,1

%A _Bernard Schott_ and _Robert G. Wilson v_, Apr 25 2022