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

Author

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

Keywords

• nonn
• base

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.

Examples

			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.
		

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.

Crossrefs

Cf. A061836 (multiplication instead of concatenation), A332580, A332585.

Programs

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;

PARI

apply( {A332584(n,L=10^#Str(n),c=n)= until((c=c*L+n)%(n+1)==0, n++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

Formula

a(n) = n + A332580(n) (trivially from the definitions).

Extensions

a(44) onwards (using A332580) added by Andrew Howroyd, Jan 02 2024