OFFSET
1,2
COMMENTS
The first three primes 2, 3 and 5 are particular cases, cf. examples. It happens that all other primes < 47 are in A180346 (and therefore have a(n) < 1000). P=37 is the only one among them with a(n) < 100 (but m=123 is another possibility for this prime).
Conjecture: a(n) > 0 for n <> 1 and n <> 3. - Chai Wah Wu, Oct 06 2023
Least m>0 such that prime(n) divides both A007908(m) and 10^A058183(m)-1; or 0 if no such m exists. - Chai Wah Wu, Oct 07 2023
LINKS
Chai Wah Wu, Table of n, a(n) for n = 1..4042
EXAMPLE
For prime(1)=2, no such m can exist (consider e.g. the initial 1 is permuted to the end), therefore a(1)=0.
For prime(2)=3, we have S(2)=12 and the permutation 21 both divisible by 3, thus a(2)=2. (There are many m for which the divisibility property is satisfied; it is equivalent to 1+...+m=0 (mod 3), or equivalently the sum of all these digits is divisible by 3. Therefore, the permutations do not need to be checked.)
For prime(3)=5, similar to prime(1)=2, no such m can exist.
For prime(4)=7, it turns out the m=100 is the least possibility, i.e., 123...99100 and the permutations 234...991001, 345...9910012, ... 100123...99, (00)123...991, (0)123...9910 are all divisible by 7.
PROG
(PARI) A181373(p, LIM=999, MIN=1)={ p=prime(p); p!=2 & p!=5 & for(n=MIN, LIM, my(S=eval(concat(vector(n, i, Str(i)))), L=#Str(S)-1); S%p & next; for(k=1, L, (S=[1, 10^L]*divrem(S, 10)) % p & next(2)); return(n)) } /* highly unoptimized code, for illustration purpose */
(Python)
from sympy import prime
def A181373(n):
s, p, l = '', prime(n), 0
for m in range(1, 10**6):
u = str(m)
s += u
l += len(u)
t = s
if not int(t) % p:
for i in range(l-1):
t = t[1:]+t[0]
if int(t) % p:
break
else:
return m
else:
return 'search limit reached.' # Chai Wah Wu, Nov 12 2015
(Python)
from itertools import count
from sympy import prime
def A181373(n):
if n == 1 or n == 3: return 0
p, c, q, a, b = prime(n), 0, 1, 10, 10
for m in count(1):
if m >= b:
a = 10*a%p
b *= 10
c = (c*a + m) % p
q = q*a % p
if not (c or (q-1)%p):
return m # Chai Wah Wu, Oct 07 2023
CROSSREFS
KEYWORD
nonn,base
AUTHOR
M. F. Hasler and Marco Ripà, Jan 27 2011
EXTENSIONS
a(15)-a(31) from Chai Wah Wu, Nov 12 2015
a(32)-a(46) from Chai Wah Wu, Oct 06 2023
STATUS
approved