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A034694 Smallest prime == 1 (mod n). 53
2, 3, 7, 5, 11, 7, 29, 17, 19, 11, 23, 13, 53, 29, 31, 17, 103, 19, 191, 41, 43, 23, 47, 73, 101, 53, 109, 29, 59, 31, 311, 97, 67, 103, 71, 37, 149, 191, 79, 41, 83, 43, 173, 89, 181, 47, 283, 97, 197, 101, 103, 53, 107, 109, 331, 113, 229, 59, 709, 61, 367, 311 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Thangadurai and Vatwani prove that a(n) <= 2^(phi(n)+1)-1. - T. D. Noe, Oct 12 2011

Conjecture: a(n) < n^2 for n > 1. - Thomas Ordowski, Dec 19 2016

REFERENCES

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 127-130.

P. Ribenboim, The Book of Prime Number Records. Chapter 4,IV.B.: The Smallest Prime In Arithmetic Progressions, 1989, pp. 217-223.

LINKS

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

S. R. Finch, More about Linnik's Constant

D. Graham, On Linnik's Constant, Acta Arithm., 39, 1981, pp. 163-179.

I. Niven and B. Powell, Primes in Certain Arithmetic Progressions, Amer. Math. Monthly, 83, 1976, pp. 467-489.

R. Thangadurai and A. Vatwani, The least prime congruent to one modulo n, Amer. Math. Monthly, Vol. 118, 2011, p. 737-742.

FORMULA

a(n) = min{m: m=k*n+1 with k>0 and A010051(m)=1}. - Reinhard Zumkeller, Dec 17 2013

EXAMPLE

If n = 7, the smallest prime in the sequence 8,15,22,29,... is 29, so a(7) = 29.

MATHEMATICA

f[n_] := Block[{k = 1}, If[n == 1, 2, While[Mod[Prime@k, n] != 1, k++ ]; Prime@k]]; Array[f, 64] (* Robert G. Wilson v, Jul 08 2006 *)

With[{prs=Prime[Range[200]]}, Join[{2}, Flatten[Table[Select[prs, Mod[#, n] == 1&, 1], {n, 2, 70}]]]] (* Harvey P. Dale, Mar 16 2012 *)

PROG

(PARI) a(n)=if(n<0, 0, s=1; while((prime(s)-1)%n>0, s++); prime(s))

(Haskell)

a034694 n = until ((== 1) . a010051) (+ n) (n + 1)

-- Reinhard Zumkeller, Dec 17 2013

CROSSREFS

Cf. A034693, A034780, A034782, A034783, A034784, A034785, A034846, A034847, A034848, A034849, A038700.

Cf. A085420. Records: A120856, A120857.

Sequence in context: A085107 A241082 A219789 * A050921 A087386 A110359

Adjacent sequences:  A034691 A034692 A034693 * A034695 A034696 A034697

KEYWORD

nonn,nice,easy

AUTHOR

Labos Elemer, David W. Wilson, Spring 1998

STATUS

approved

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Last modified May 26 05:25 EDT 2017. Contains 287077 sequences.