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A035095
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Smallest prime congruent to 1 (mod prime(n)).
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21
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3, 7, 11, 29, 23, 53, 103, 191, 47, 59, 311, 149, 83, 173, 283, 107, 709, 367, 269, 569, 293, 317, 167, 179, 389, 607, 619, 643, 1091, 227, 509, 263, 823, 557, 1193, 907, 1571, 653, 2339, 347, 359, 1087, 383, 773, 3547, 797, 2111, 2677, 5449, 2749, 467
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OFFSET
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1,1
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COMMENTS
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This is a version of the "least prime in special arithmetic progressions" problem.
Smallest numbers m such that largest prime factor of Phi(m) = prime(n), the n-th prime, also seems to be prime and identical to n-th term of A035095. See A068211, A068212, A065966: Min[x : A068211(x)=prime(n)] = A035095(n); e.g., Phi(a(7)) = Phi(103) = 2*3*17, of which 17 = p(7) is the largest prime factor, arising first here.
Minimum of the smallest prime factors of F(n,i) = (i^prime(n)-1)/(i-1), when i runs through all integers in [2, prime(n)]. Every prime factor of F(n,i) is congruent to 1 modulo prime(n). - Vladimir Shevelev, Nov 26 2014
Conjecture: a(n) is the smallest prime p such that gpf(p-1) = prime(n). See A023503. - Thomas Ordowski, Aug 06 2017
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REFERENCES
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E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Bd 1 (reprinted Chelsea 1953).
E. C. Titchmarsh, A divisor problem, Renc. Circ. Math. Palermo, 54 (1930) pp. 414-429.
P. Turan, Uber Primzahlen der arithmetischen Progression, Acta Sci. Math. (Szeged), 8 (1936/37) pp. 226-235.
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LINKS
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U. V. Linnik, On the least prime in an arithmetic progression. I. The basic theorem, Rec. Math (N.S.) v 15 (57) (1944), pp 139-178. MR0012111
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FORMULA
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According to a long-standing conjecture (see the 1979 Wagstaff reference), a(n) <= prime(n)^2 + 1. This would be sufficient to imply that a(n) is the smallest prime such that greatest prime divisor of a(n)-1 is prime(n), the n-th prime: A006530(a(n)-1) = A000040(n). This in turn would be sufficient to imply that no value occurs twice in this sequence. - Franklin T. Adams-Watters, Jun 18 2010
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EXAMPLE
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a(8) = 191 because in the prime(8)k+1 = 19k+1 sequence, 191 is the smallest prime.
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MATHEMATICA
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a[n_] := Block[{p = Prime[n]}, r = 1 + p; While[ !PrimeQ[r], r += p]; r]; Array[a, 51] (* Jean-François Alcover, Sep 20 2011, after PARI *)
a[n_]:=If[n<2, 3, Block[{p=Prime[n]}, r=1+2*p; While[!PrimeQ[r], r+=2*p]]; r]; Array[a, 51] (* Zak Seidov, Dec 14 2013 *)
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PROG
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(PARI) a(n)=local(p, r); p=prime(n); r=1; while(!isprime(r), r+=p); r
(PARI) {my(N=66); forprime(p=2, , forprime(q=p+1, 10^10, if((q-1)%p==0, print1(q, ", "); N-=1; break)); if(N==0, break)); } \\ Joerg Arndt, May 27 2016
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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