A332867 a(n) = minimal positive k such that the concatenation of decimal digits 1,2,...,n is a divisor of the concatenation of n+1,n+2,...,n+k.

1, 4, 20, 1834, 14995, 6986, 2888370, 795412, 37366487, 8803255100

Offset: 1

Author

Scott R. Shannon, Feb 27 2020

Keywords

• nonn
• base
• more
• hard

Comments

As with A332580 a heuristic argument, based on the divergent sum of reciprocals which approximates the probability that the concatenation of 1,2,...,n will divide the concatenation of n+1,n+2,...,n+k suggests that k should always exist. However an examination of the prime factors of the concatenation of 1,2,...,n shows that most of these numbers contain one or more very large primes, suggesting the values of k will likely become extremely large as n increases.

The author thanks Joseph Myers for suggestions for finding the larger terms of this sequence.

Examples

			a(2) = 4 as '1'||'2' = 12 and '3'||'4'||'5'||'6' = 3456, which is divisible by 12 (where '||' denotes decimal concatenation).
a(3) = 20 as '1'||'2'||'3' = 123 and '4'||'5'||...||'22'||'23' = 4567891011121314151617181920212223, which is divisible by 123.
		

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. A332580, A332830, A007908.

Programs

Maple

a:= proc(n) local i, t, m; t, m:= parse(cat($1..n)), 0;
      for i from n+1 do m:= parse(cat(m,i)) mod t;
      if m=0 then break fi od; i-n
    end:
seq(a(n), n=1..6);  # Alois P. Heinz, Feb 29 2020

PARI

a(n) = {my(k=1, small="", big = n+1); for (j=1, n, small=concat(small, Str(j))); small = eval(small); while (big % small, k++; big = eval(concat(Str(big), Str(n+k)))); k;} \\ Michel Marcus, Feb 29 2020

Python

def A332867(n):
    m, k = int(''.join(str(d) for d in range(1,n+1))), 1
    i = n+k
    i2, l = i % m, len(str(i))
    t = 10**l
    t2, r = t % m, i % m
    while r != 0:
        k += 1
        i += 1
        i2 = (i2+1) % m
        if i >= t:
            l += 1
            t *= 10
            t2 = (10*t2) % m
        r = (r*t2 + i2) % m
    return k # Chai Wah Wu, May 20 2020