

A332584


a(n) = minimal value of n+k (with k >= 1) such that the concatenation of the decimal digits of n,n+1,...,n+k is divisible by n+k+1, or 1 if no such n+k exists.


7



2, 82, 1888, 6842, 6, 50, 20, 10, 1320, 28, 208, 32, 66, 148, 1008, 60, 192, 124536, 282, 46, 128, 32, 28, 86, 40, 33198, 36, 42, 346, 738, 1532, 246, 70, 68, 102, 306, 56, 20226, 78316, 10778, 328, 2432, 738
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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 extendedtable of values.
A heuristic argument suggests that n+k should always exist.


LINKS

Table of n, a(n) for n=1..43.
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++<LL*=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

Cf. A061836 (multiplication instead of concatenation), A332580, A332585.
Sequence in context: A202965 A307583 A061994 * A197641 A093666 A246002
Adjacent sequences: A332581 A332582 A332583 * A332585 A332586 A332587


KEYWORD

nonn,base,more


AUTHOR

Scott R. Shannon and N. J. A. Sloane, Feb 16 2020


STATUS

approved



