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A034694 Smallest prime == 1 (mod n). 61
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
Eric Bach and Jonathan Sorenson show that, assuming GRH, a(n) <= (1 + o(1))*(phi(n)*log(n))^2 for n > 1. See the abstract of their paper in the Links section. - Jianing Song, Nov 10 2019
a(n) is the smallest prime p such that the multiplicative group modulo p has a subgroup of order n. - Joerg Arndt, Oct 18 2020
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge, 2003, section 2.12, 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
Eric Bach and Jonathan Sorenson, Explicit bounds for primes in residue classes, Mathematics of Computation, 65(216) (1996), 1717-1735.
Steven R. Finch, Linnik's Constant
S. 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(6) (1976), 467-469.
R. Thangadurai and A. Vatwani, The least prime congruent to one modulo n, Amer. Math. Monthly 118(8) (2011), 737-742.
FORMULA
a(n) = min{m: m = k*n + 1 with k > 0 and A010051(m) = 1}. - Reinhard Zumkeller, Dec 17 2013
a(n) = n * A034693(n) + 1. - Joerg Arndt, Oct 18 2020
EXAMPLE
If n = 7, the smallest prime in the sequence 8, 15, 22, 29, ... is 29, so a(7) = 29.
MATHEMATICA
a[n_] := Block[{k = 1}, If[n == 1, 2, While[Mod[Prime@k, n] != 1, k++ ]; Prime@k]]; Array[a, 64] (* Robert G. Wilson v, Jul 08 2006 *)
With[{prs=Prime[Range[200]]}, Flatten[Table[Select[prs, Mod[#-1, n]==0&, 1], {n, 70}]]] (* Harvey P. Dale, Sep 22 2021 *)
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
Records: A120856, A120857.
Sequence in context: A085107 A241082 A219789 * A050921 A087386 A110359
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved

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Last modified March 19 06:56 EDT 2024. Contains 370953 sequences. (Running on oeis4.)