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A344734
a(n) is the smallest divisibility checking number for the n-th number coprime to 10.
0
1, 2, 2, 8, 1, 9, 5, 17, 2, 16, 8, 26, 3, 23, 11, 35, 4, 30, 14, 44, 5, 37, 17, 53, 6, 44, 20, 62, 7, 51, 23, 71, 8, 58, 26, 80, 9, 65, 29, 89, 10, 72, 32, 98, 11, 79, 35, 107, 12, 86, 38, 116, 13, 93, 41, 125, 14, 100, 44, 134, 15, 107, 47, 143, 16, 114, 50
OFFSET
1,2
COMMENTS
Conjecture: the subsequence of divisibility checking numbers for numbers that end in 1 form an arithmetic sequence. This is also the case for numbers ending in 3, 7, and 9.
FORMULA
Conjectures from Chai Wah Wu, Dec 29 2021: (Start)
a(n) = 2*a(n-4) - a(n-8) for n > 9.
G.f.: x*(x^8 + x^7 + x^6 + 5*x^5 - x^4 + 8*x^3 + 2*x^2 + 2*x + 1)/(x^8 - 2*x^4 + 1). (End)
EXAMPLE
For n = 6, the sixth number coprime to 10 is 13. The divisibility checking number for 13 is 9, as demonstrated below:
13 divides 12961
1296 - (1 * 9) = 1287
13 divides 1287
128 - (7 * 9) = 65
13 divides 65.
This is useful in quickly checking divisibility by hand. 13 divides 65 implies that 13 divides 12961.
PROG
(Python)
def divisibility_checking_number(a):
if a%5 == 0 or a % 2 == 0:
return(0)
x = 1
while (x*10 + 1)%a != 0 :
x += 1
return(x)
h = []
for i in range(1, 250):
if divisibility_checking_number(i) != 0:
h.insert(len(h), divisibility_checking_number(i))
print(h)
CROSSREFS
Cf. A045572 (numbers coprime to 10).
Sequence in context: A102645 A366324 A365747 * A037300 A368877 A029623
KEYWORD
nonn
AUTHOR
Zach Pollard, May 27 2021
STATUS
approved