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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. 4
1, 4, 20, 1834, 14995, 6986, 2888370, 795412, 37366487, 8803255100 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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.
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.
EXAMPLE
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.
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
PROG
(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
CROSSREFS
Sequence in context: A144989 A012841 A074195 * A217369 A146342 A103896
KEYWORD
nonn,base,more,hard
AUTHOR
Scott R. Shannon, Feb 27 2020
STATUS
approved

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Last modified July 24 10:37 EDT 2024. Contains 374583 sequences. (Running on oeis4.)