A332830 a(n) = minimal positive k such that the concatenation of decimal digits n and n+1 is a divisor of the concatenation of n+2, n+2+1, ..., n+2+k.

3, 4, 3, 24, 13, 7, 33, 7, 749, 125, 1019, 3643, 123, 1319, 1199, 1424, 1481, 664, 659, 734, 6139, 933, 607, 549, 165, 8124, 63, 296, 1339, 13817, 1691, 6979, 3, 704, 2187, 156, 987, 2521, 1459, 1277, 6047, 25565, 3179, 1954, 7127, 1115, 6139, 18749, 1149

Offset: 1

Author

Scott R. Shannon, Feb 25 2020

Keywords

• nonn
• base

Comments

Like A332580 a heuristic argument, based on the divergent sum of reciprocals which approximates the probability that the concatenation of n and n+1 will divide the concatenation of n+2, n+3, ..., suggests that k should always exist.

Examples

			a(1) = 3 as '1'||'2' = 12 and '3'||'4'||'5'||'6' = 3456, which is divisible by 12 (where '||' denotes decimal concatenation).
a(4) = 24 as '4'||'5' = 45 and '6'||'7'||....||'29'||'30' = 6789101112131415161718192021222324252627282930, which is divisible by 45.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A332580, A332867, A007908.

Programs

Maple

a:= proc(n) local i, t, m; t, m:= parse(cat(n,n+1)), 0;
      for i from n+2 do m:= parse(cat(m,i)) mod t;
      if m=0 then break fi od; i-n-2
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, Feb 29 2020

PARI

a(n) = {my(k=1, small=eval(concat(Str(n), Str(n+1))), big=n+2); while( big % small, big = eval(concat(Str(big), Str(n+2+k))); k++); k--;} \\ Michel Marcus, Feb 29 2020