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A035095 Smallest prime congruent to 1 mod n-th prime. 16
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

This is a version of the "least prime in special arithmetic progressions" problem.

Smallest numbers m such that largest prime factor of Phi(m) = p(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)=p(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.

It appears that A035095, A066674, A125878 are probably all the same, but see the comments in A066674. - N. J. A. Sloane, Jan 05 2013

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

REFERENCES

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.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

P. Erdős, On some application of Brun's method, Acta Sci. Math (Szeged), v. 13, 1949 pp 57-63

A. Granville and C. Pomerance, On the least prime in certain arithmetic progressions

A. Granville and C. Pomerance, On the least prime in certain arithmetic progressions J. Lond Math Soc s2-41 (2) (1990) 193-200

D. R. Heath-Brown, almost-primes in arithmetic progressions in short intervals, Math Proc Cambr. Phil Soc v 83 (1978) pp 357-375

D. R. Heath-Brown, Siegel zeros and the least prime in arithmetic progression, Quart. J. of Math 41 (49) (1990) 405-418

H.-J. Kanold, Uber Primzahlen in arithmetischen Folgen, Math. Ann. v 156 (1964) pp 393-395

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

C. Pomerance, A note on the least prime in an arithmetic progression J. Number Theory 12 (2) (1980) 218-223

K. Prachar, Uber die kleinste Primzahl in einer arithmetischen Reihe, J Reine Angew Math. 206 (1961) pp. 3-4

A. Schinzel, Remark on the paper of K. Prachar Uber die kleinste.., J. Reine Angew Math. v 210 (1962) pp 122-122

S. S. Wagstaff, Jr., The irregular primes to 125000, Math. Comp., 32 (1978) pp. 583-591.

S. S. Wagstaff, Jr, Greatest of the Least Primes in Arithmetic Progressions Having a Given Modulus, Math. Comp., 33 (147) (1979) pp. 1073-1080

Index entries for sequences related to primes in arithmetic progressions

FORMULA

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

a(n) = 1 + A035096(n)*A000040(n). - Zak Seidov, Dec 27 2013

EXAMPLE

a(8) = 191 because in the prime(8)k+1 = 19k+1 sequence, 191 is the smallest prime.

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

Cf. A034694, A032448, A006530, A006093, A035096, A000040, A019434, A058383.

Cf. A068211, A068212, A065966, A000010, A070844-A070858, A061092.

Cf. A000040.

Sequence in context: A211674 * A066674 A125878 A126112 A194373 A156210

Adjacent sequences:  A035092 A035093 A035094 * A035096 A035097 A035098

KEYWORD

nonn

AUTHOR

Labos Elemer

EXTENSIONS

Edited by Franklin T. Adams-Watters, Jun 18 2010

Minor edits by N. J. A. Sloane, Jun 27 2010

Edited by N. J. A. Sloane, Jan 05 2013

STATUS

approved

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Last modified October 22 02:06 EDT 2017. Contains 293756 sequences.