A180346 Primes that divide every circular permutation of the digits of at least one number of the form 123...(n-1)(n) (see A007908), where n is 3 digits long (that is, for some n in the range 99

3, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 53, 61, 67, 73, 83, 97, 101, 107, 127, 163, 211, 271, 277, 1009, 18973

Offset: 1

Author

Marco Ripà, Jan 22 2011

Keywords

• base
• fini
• hard
• nonn

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

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 99Chai 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.

Crossrefs

Cf. A007908, A058183, A181373.

Programs

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
A180346_list = list(islice(A180346_gen(),20)) # Chai Wah Wu, Oct 07 2023

Formula

For n<10 the only a(i) is 3. If 9

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