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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

%I A180346 #55 Oct 10 2023 05:32:54
%S A180346 3,7,11,13,17,19,23,29,31,37,41,43,53,61,67,73,83,97,101,107,127,163,
%T A180346 211,271,277,1009,18973
%N 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<n<1000).
%C A180346 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]:
%C A180346 a(2) | 7  ⇒  n=100+14*v       (v=0,1,2,...,64)
%C A180346 a(3) | 11 ⇒  n=106+22*v       (v=0,1,2,...,40)
%C A180346 a(4) | 13 ⇒  n=120+26*v       (v=0,1,2,...,33)
%C A180346 a(5) | 17 ⇒  n=196+272*v      (v=0,1,2)
%C A180346 a(6) | 19 ⇒  n=102+114*v      (v=0,1,2,3,4,5,6,7)
%C A180346 a(7) | 23 ⇒  n=542
%C A180346 a(8) | 29 ⇒  n=400
%C A180346 a(9) | 31 ⇒  n=181+155*v      (v=0,1,2,3,4,5)
%C A180346 a(10)| 37 ⇒  n=123+d(v),
%C A180346 (where d(v)=0,12,25,12,25,12,25...  for v=0,1,2,3,...,47)
%C A180346 a(11) | 41 ⇒  n=216+205*v     (v=0,1,2,3)
%C A180346 a(12) | 43 ⇒  n=372+301*v     (v=0,1,2)
%C A180346 a(13) | 53 ⇒  n=127+689*v     (v=0,1)
%C A180346 a(14) | 61 ⇒  n=616
%C A180346 a(15) | 67 ⇒  n=399
%C A180346 a(16) | 73 ⇒  n=196+584*v     (v=0,1)
%C A180346 a(17) | 83 ⇒  n=118
%C A180346 a(18) | 97 ⇒  n=516
%C A180346 a(19) | 101 ⇒  n=416+404*v    (v=0,1)
%C A180346 a(20) | 107 ⇒  n=884
%C A180346 a(21) | 127 ⇒  n=106
%C A180346 a(22) | 163 ⇒  n=576
%C A180346 a(23) | 211 ⇒  n=306
%C A180346 a(24) | 271 ⇒  n=936
%C A180346 a(25) | 277 ⇒  n=174
%C A180346 a(26) | 1009 ⇒ n=960
%C A180346 a(27) | 18973 ⇒ n=903
%C A180346 N.B.
%C A180346 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).
%C A180346 a(28) >= prime(10^6) if it exists. - _Chai Wah Wu_, Nov 12 2015
%C A180346 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
%C A180346 a(28) > prime(2.3316*10^9) if it exists. Conjecture: 18973 is the last term. - _Chai Wah Wu_, Oct 09 2023
%D A180346 Vassilev-Missana and K. Atanassov, “Some Smarandache problems”, Hexis, 2004.
%H A180346 Marco Ripà, <a href="http://www.rudimathematici.com/bookshelf/pdf/PrimiRipa.pdf">Strutture modulari associate al problema della primalità di alcune sequenze concatenate</a>, Rudimatematici, Bookshelf, October 2010. In Italian.
%H A180346 Marco Ripà, <a href="http://vixra.org/abs/1101.0092">On prime factors in old and new sequences of integers</a>, vixra, 2011.
%H A180346 Marco Ripa, <a href="http://www.nntdm.net/papers/nntdm-18/NNTDM-18-1-29-48.pdf">Patterns related to the Smarandache circular sequence primality problem</a>, Notes Numb. Th. Discr. Math., vol. 18(1) (2012), pp. 29-48.
%H A180346 F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/OPNS.pdf">Only Problems, Not Solutions!</a>, Xiquan Publ., Phoenix-Chicago, 1993.
%F A180346 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).
%o A180346 (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
%o A180346 (Python)
%o A180346 from itertools import islice
%o A180346 from sympy import nextprime
%o A180346 def A180346_gen(startvalue=1): # generator of terms >= startvalue
%o A180346     p = max(startvalue-1,0)
%o A180346     while (p:=nextprime(p)):
%o A180346         c, q, a, b = 0, 1, 10, 10
%o A180346         for m in range(1,1000):
%o A180346             if m >= b:
%o A180346                 a = 10*a%p
%o A180346                 b *= 10
%o A180346             c = (c*a + m) % p
%o A180346             q = q*a % p
%o A180346             if m>99 and not (c or (q-1)%p):
%o A180346                 yield p
%o A180346                 break
%o A180346 A180346_list = list(islice(A180346_gen(),20)) # _Chai Wah Wu_, Oct 07 2023
%Y A180346 Cf. A007908, A058183, A181373.
%K A180346 base,fini,hard,nonn
%O A180346 1,1
%A A180346 _Marco Ripà_, Jan 22 2011