A034780 -id:A034780 - cp's OEIS frontend

A034847 a(n) = 1 + 4*A034780(n).

29, 53, 101, 109, 149, 173, 181, 197, 229, 269, 293, 317, 337, 349, 373, 389, 461, 509, 557, 569, 641, 653, 677, 701, 709, 773, 797, 821, 829, 853, 937, 941, 1013, 1021, 1033, 1061, 1069, 1109, 1117, 1181, 1193, 1217, 1229, 1277, 1297, 1301, 1373, 1429, 1481, 1493, 1549, 1597

Offset: 1

Labos Elemer

nonn

a(n) = P(n,4) = 1 + 4*K(n,4) = 1 + 4*A034780(n). P(n,4) are special primes of the form 4k+1. The relevant values of k are given by A034780.

Note that, e.g., 5 and 13 are not in this sequence.

Cf. A034693, A034694, A034780.

a034693(n) = my(s=1); while(!isprime(s*n+1), s++); s;
isok(n) = a034693(n) == 4;
lista(nn) = {for (n=1, nn, if (isok(n), print1(4*n+1, ", ")););} \\ Michel Marcus, May 13 2018

More terms from Michel Marcus, May 13 2018

A034694 Smallest prime == 1 (mod n).

Original entry on oeis.org

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

Offset: 1

Labos Elemer, David W. Wilson, Spring 1998

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

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

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.

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

Cf. A034693, A034780, A034782, A034783, A034784, A034785, A034846, A034847, A034848, A034849, A038700, A085420.

Records: A120856, A120857.

a034694 n = until ((== 1) . a010051) (+ n) (n + 1)
-- Reinhard Zumkeller, Dec 17 2013

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 *)

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

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

A047980 a(n) is smallest difference d of an arithmetic progression dk+1 whose first prime occurs at the n-th position.

Original entry on oeis.org

1, 3, 24, 7, 38, 17, 184, 71, 368, 19, 668, 59, 634, 167, 512, 757, 1028, 197, 1468, 159, 3382, 799, 4106, 227, 10012, 317, 7628, 415, 11282, 361, 38032, 521, 53630, 3289, 37274, 2633, 63334, 1637, 34108, 1861, 102296, 1691, 119074, 1997, 109474, 2053

Offset: 1

Labos Elemer

nonn

Definition involves two minimal conditions: (1) the first prime (as in A034693) and (2) dk+1 sequences were searched with minimal d. Present terms are the first ones in sequences analogous to A034780, A034782-A034784, A006093 (called there K(n,m)).

Index of the first occurrence of n in A034693. - Amarnath Murthy, May 08 2003

			For n=2, the sequence with d=1 is 2,3,4,5,... with the prime 2 for k=1.  The sequence with d=2 is 3,5,7,9,... with the prime 3 for k=1.  The sequence with d=3 is 4,7,10,13,... with the prime 7 for k=2.  So a(n)=3. - _Michael B. Porter_, Mar 18 2019

Jon E. Schoenfield, Table of n, a(n) for n = 1..150 (terms 1..72 from Robert Israel)
Jon E. Schoenfield, Terms <= 5*10^8: Table of n, a(n) for n = 1..406, with -1 for each term > 5*10^8
Index entries for sequences related to primes in arithmetic progressions

Cf. A034693, A034694, A034780, A034782, A034783, A034784, A006093, A047981, A047982.

function [ A ] = A047980( P, N )
%   Get values a(i) for i <= N with a(i) <= P/i
%   using primes <= P.
%   Returned entries A(n) = 0 correspond to unknown a(n) > P/n
Primes = primes(P);
A = zeros(1,N);
Ds = zeros(1,P);
for p = Primes
   ns = [1:N];
   ns = ns(mod((p-1) * ones(1,N), ns) == 0);
   newds = (p-1) ./ns;
   ns = ns(A(ns) == 0);
   ds = (p-1) ./ ns;
   q = (Ds(ds) == 0);
   A(ns(q)) = ds(q);
   Ds(newds) = 1;
end
end % Robert Israel, Jan 25 2016

N:= 40: # to get a(n) for n <= N
count:= 0:
p:= 0:
Ds:= {1}:
while count < N do
    p:= nextprime(p);
    ds:= select(d -> (p-1)/d <= N, numtheory:-divisors(p-1) minus Ds);
    for d in ds do
      n:= (p-1)/d;
      if not assigned(A[n]) then
        A[n]:= d;
        count:= count+1;
      fi
    od:
    Ds:= Ds union ds;
od:
seq(A[i],i=1..N); # Robert Israel, Jan 25 2016

With[{s = Table[k = 1; While[! PrimeQ[k n + 1], k++]; k, {n, 10^6}]}, TakeWhile[#, # > 0 &] &@ Flatten@ Array[FirstPosition[s, #] /. k_ /; MissingQ@ k -> {0} &, Max@ s]] (* Michael De Vlieger, Aug 01 2017 *)

a(n) = min{k | A034693(k) = n}.

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A034847 a(n) = 1 + 4*A034780(n).

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A034694 Smallest prime == 1 (mod n).

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A047980 a(n) is smallest difference d of an arithmetic progression dk+1 whose first prime occurs at the n-th position.

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