cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A034848 a(n) = 1 + 3*A034782(n).

Original entry on oeis.org

73, 97, 103, 193, 229, 241, 277, 283, 313, 331, 367, 373, 397, 433, 457, 463, 547, 607, 619, 643, 661, 709, 727, 733, 739, 757, 823, 859, 883, 907, 967, 997, 1021, 1033, 1069, 1087, 1093, 1123, 1129, 1171, 1237, 1249, 1303, 1423, 1447, 1453, 1483, 1489, 1543, 1579, 1597
Offset: 1

Views

Author

Labos Elemer

Keywords

Comments

a(n) = P(n,3) = 1 + 3*K(n,3) = 1 + 3*A034782(n). P(n,3) are special primes of the form 3k+1. The relevant values of k are given by A034782.
Note that, e.g., 13, 19, 31, 5, 13 are not in this sequence.

Crossrefs

Cf. A034693, A034694, A034782.

Programs

  • PARI
    a034693(n) = my(s=1); while(!isprime(s*n+1), s++); s;
isok(n) = a034693(n) == 3;
lista(nn) = {for (n=1, nn, if (isok(n), print1(3*n+1, ", ")););} \\ Michel Marcus, May 13 2018

Extensions

Corrected (wrong term 769 removed) and extended by 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

Views

Author

Labos Elemer, David W. Wilson, Spring 1998

Keywords

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

Examples

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

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

Crossrefs

Cf. A034693, A034780, A034782, A034783, A034784, A034785, A034846, A034847, A034848, A034849, A038700, A085420.
Records: A120856, A120857.

Programs

  • Haskell
    a034694 n = until ((== 1) . a010051) (+ n) (n + 1)
-- Reinhard Zumkeller, Dec 17 2013
  • 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 *)
  • PARI
    a(n)=if(n<0,0,s=1; while((prime(s)-1)%n>0,s++); prime(s))

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

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

Views

Author

Labos Elemer

Keywords

Comments

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

Examples

			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

Links

Crossrefs

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

Programs

  • MATLAB
    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
  • Maple
    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
  • Mathematica
    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 *)

Formula

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

A047981 a(n) = A047980(2n).

Original entry on oeis.org

3, 7, 17, 71, 19, 59, 167, 757, 197, 159, 799, 227, 317, 415, 361, 521, 3289, 2633, 1637, 1861, 1691, 1997, 2053, 4097, 6437, 5731, 9199, 11603, 5641, 3833, 26885, 6637, 26815, 32117, 18637, 29933, 31667, 5227, 19891, 47303, 54973, 5207, 59537
Offset: 1

Views

Author

Labos Elemer

Keywords

Examples

			First example: a(1)=3 since in 3k+1 sequence, the first term is 3, a prime and the d=2 is the smallest such difference. The next such progression is 5k+1 because 5*2+1=11 is prime. 2nd example: here at n=6 a(6)=59. This means that 2n=12 occurs first in A034693 at its position 59, which means that its first prime is 12*59+1=709. arises as 12th term (such progressions are: 59k+1,85k+1,133k+1, etc.)

Links

Crossrefs

Cf. A047980, A047982, A034693, A034782 - A034784.

Formula

a(n) = min {k}: A034693(a(n)) is an even number such that in a(n)*k+1 progression the first prime occurs at even 2n=k position.

A047982 a(n) = A047980(2n+1).

Original entry on oeis.org

1, 24, 38, 184, 368, 668, 634, 512, 1028, 1468, 3382, 4106, 10012, 7628, 11282, 38032, 53630, 37274, 63334, 34108, 102296, 119074, 109474, 117206, 60664, 410942, 204614, 127942, 125618, 595358, 517882, 304702, 352022, 1549498, 651034, 506732, 5573116, 1379216, 1763144
Offset: 0

Views

Author

Labos Elemer

Keywords

Examples

			a(2)=38 because A034693(38) = 2*2+1 = 5 is the first 5; 5*38+1 = 191 is the first prime. The successive progressions in which the first prime appears at position 5 are as follows: 38k+1, 62k+1, 164k+1. 2nd example: a(20)=102296 because. The first 41 appears in A034693 at this index. Also 102296*(2*20+1)+1 = 102296*41+1 = 4194137 is the first prime in {102296k+1}. The next progression with this position of prime emergence is 109946k+1 (the corresponding prime is 4507787).

Links

Crossrefs

Cf. A047980, A047981, A034782, A034783, A034784.

Formula

a(n) = min {d}: A034693(a(n)) is an odd number k such that in a(n)*k+1 progression the first prime occurs at k=2n+1 position.

Extensions

More terms from Michel Marcus, Sep 01 2019
Showing 1-5 of 5 results.

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