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.

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, 2783191412956, 48, 746, 8350, 398, 70, 150, 2300, 21378

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

Table of n, a(n) for n=1..52.

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

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

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