OFFSET
1,1
COMMENTS
Certainly a(n) must be even, since no odd number can be divisible by an even number.
The values of k = a(n)-n are given in the companion sequence A332580, which also has an extended table of values.
A heuristic argument suggests that n+k should always exist.
LINKS
J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000 [math.NT], April 2020.
FORMULA
a(n) = n + A332580(n) (trivially from the definitions).
EXAMPLE
a(1) = 2 as '1' || '2' = '12', which is divisible by 3 (where || denotes decimal concatenation).
a(7) = 20 as '7' || '8' || '9' || '10' || '11' || '12' || ... || '20' = 7891011121314151617181920, which is divisible by 21.
a(8) = 10 as '8' || '9' || '10' = 8910, which is divisible by 11.
a(2) = 82: the concatenation 2 || 3 || ... || 82 is
23456789101112131415161718192021222324252627282930313233343536373839\
40414243444546474849505152535455565758596061626364656667686970717273747\
576777879808182, which is divisible by 83.
MAPLE
grow := proc(n, M) # searches out to a limit of M, returns [n, n+k] or [n, -1] if no k was found
local R, i;
R:=n;
for i from n+1 to M do
R:=R*10^length(i)+i;
if (i mod 2) = 0 then
if (R mod (i+1)) = 0 then return([n, i]); fi;
fi;
od:
[n, -1];
end;
for n from 1 to 100 do lprint(grow(n, 20000)); od;
PROG
(PARI) apply( {A332584(n, L=10^#Str(n), c=n)= until((c=c*L+n)%(n+1)==0, n++<L||L*=10); n}, [1..17]) \\ M. F. Hasler, Feb 20 2020
(Python)
def A332584(n):
r, m = n, n + 1
while True:
r = r*10**(len(str(m))) + m
if m % 2 == 0 and r % (m+1) == 0:
return m
m += 1 # Chai Wah Wu, Jun 12 2020
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Scott R. Shannon and N. J. A. Sloane, Feb 16 2020
EXTENSIONS
a(44) onwards (using A332580) added by Andrew Howroyd, Jan 02 2024
STATUS
approved