

A181373


Least m>0 such that prime(n) divides S(m)=A007908(m)=123...m and all numbers obtained by cyclic permutations of its digits; 0 if no such m exists.


2



0, 2, 0, 100, 106, 120, 196, 102, 542, 400, 181, 21, 216, 372, 10446, 127, 10086, 616, 399, 1703, 196, 2009, 118, 12350, 516, 416, 13244, 884, 15462, 15146, 106
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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).


LINKS

Table of n, a(n) for n=1..31.


FORMULA

A007908( A181373(n) ) = 0 (mod A000040(n)).


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(l1):
t = t[1:]+t[0]
if int(t) % p:
break
else:
return m
else:
return 'search limit reached.' # Chai Wah Wu, Nov 12 2015


CROSSREFS

Cf. A000040, A007908, A180346 (see references there), A181373.
Sequence in context: A307946 A009270 A033838 * A046065 A339185 A003321
Adjacent sequences: A181370 A181371 A181372 * A181374 A181375 A181376


KEYWORD

nonn,base,more


AUTHOR

M. F. Hasler and Marco Ripà, Jan 27 2011


EXTENSIONS

a(15)a(31) from Chai Wah Wu, Nov 12 2015


STATUS

approved



