OFFSET
1,1
COMMENTS
Every a(i) divides at least 192 permutations of the digits of an element belonging to [A007908]. Skipping the trivial case a(1)=3, the most recurring elements are a(2)=7 and a(10)=37. The occurrences in our 1386450 terms set are the following [A181373]:
a(2) | 7 ⇒ n=100+14*v (v=0,1,2,...,64)
a(3) | 11 ⇒ n=106+22*v (v=0,1,2,...,40)
a(4) | 13 ⇒ n=120+26*v (v=0,1,2,...,33)
a(5) | 17 ⇒ n=196+272*v (v=0,1,2)
a(6) | 19 ⇒ n=102+114*v (v=0,1,2,3,4,5,6,7)
a(7) | 23 ⇒ n=542
a(8) | 29 ⇒ n=400
a(9) | 31 ⇒ n=181+155*v (v=0,1,2,3,4,5)
a(10)| 37 ⇒ n=123+d(v),
(where d(v)=0,12,25,12,25,12,25... for v=0,1,2,3,...,47)
a(11) | 41 ⇒ n=216+205*v (v=0,1,2,3)
a(12) | 43 ⇒ n=372+301*v (v=0,1,2)
a(13) | 53 ⇒ n=127+689*v (v=0,1)
a(14) | 61 ⇒ n=616
a(15) | 67 ⇒ n=399
a(16) | 73 ⇒ n=196+584*v (v=0,1)
a(17) | 83 ⇒ n=118
a(18) | 97 ⇒ n=516
a(19) | 101 ⇒ n=416+404*v (v=0,1)
a(20) | 107 ⇒ n=884
a(21) | 127 ⇒ n=106
a(22) | 163 ⇒ n=576
a(23) | 211 ⇒ n=306
a(24) | 271 ⇒ n=936
a(25) | 277 ⇒ n=174
a(26) | 1009 ⇒ n=960
a(27) | 18973 ⇒ n=903
N.B.
Every coefficient of "v" is a multiple of i. This is a general property of [A007908], valid for an arbitrary fixed digits interval of the parameter "n" (10^k-1<n<10^k).
a(28) >= prime(10^6) if it exists. - Chai Wah Wu, Nov 12 2015
Primes p such that p divides both A007908(m) and 10^A058183(m)-1 for some 99<m<1000. - Chai Wah Wu, Oct 07 2023
a(28) > prime(2.3316*10^9) if it exists. Conjecture: 18973 is the last term. - Chai Wah Wu, Oct 09 2023
REFERENCES
Vassilev-Missana and K. Atanassov, “Some Smarandache problems”, Hexis, 2004.
LINKS
Marco Ripà, Strutture modulari associate al problema della primalità di alcune sequenze concatenate, Rudimatematici, Bookshelf, October 2010. In Italian.
Marco Ripà, On prime factors in old and new sequences of integers, vixra, 2011.
Marco Ripa, Patterns related to the Smarandache circular sequence primality problem, Notes Numb. Th. Discr. Math., vol. 18(1) (2012), pp. 29-48.
F. Smarandache, Only Problems, Not Solutions!, Xiquan Publ., Phoenix-Chicago, 1993.
FORMULA
For n<10 the only a(i) is 3. If 9<n<100 an a(i) is 37 (the other one is a(1)=3), which divides all the circular permutations of Sm(21).
PROG
(PARI) isA180346(p)={ isprime(p) & p!=2 & p!=5 & for(n=100, 999, 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)) } /* returns the least corresponding n or 0 if not in this sequence */ \\ M. F. Hasler, Jan 23 2011
(Python)
from itertools import islice
from sympy import nextprime
def A180346_gen(startvalue=1): # generator of terms >= startvalue
p = max(startvalue-1, 0)
while (p:=nextprime(p)):
c, q, a, b = 0, 1, 10, 10
for m in range(1, 1000):
if m >= b:
a = 10*a%p
b *= 10
c = (c*a + m) % p
q = q*a % p
if m>99 and not (c or (q-1)%p):
yield p
break
CROSSREFS
KEYWORD
base,fini,hard,nonn
AUTHOR
Marco Ripà, Jan 22 2011
STATUS
approved