A034694 -id:A034694 - cp's OEIS frontend

A034780 Numbers k such that A034693(k) = 4.

7, 13, 25, 27, 37, 43, 45, 49, 57, 67, 73, 79, 84, 87, 93, 97, 115, 127, 139, 142, 160, 163, 169, 175, 177, 193, 199, 205, 207, 213, 234, 235, 253, 255, 258, 265, 267, 277, 279, 295, 298

Offset: 1

Labos Elemer

nonn

R. J. Mathar, Table of n, a(n) for n = 1..656

Cf. A034693, A034694.

Position[#, 4] &@ Table[k = 1; While[! PrimeQ[k n + 1], k++]; k, {n, 300}] // Flatten (* Michael De Vlieger, Mar 02 2017 *)

A034783 Numbers k such that A034693(k) = 6.

Original entry on oeis.org

17, 47, 55, 61, 77, 91, 101, 103, 107, 118, 121, 123, 137, 143, 147, 151, 161, 170, 181, 187, 195, 208, 217, 237, 241, 247, 248, 257, 263, 266, 271, 283, 287, 291, 297, 305, 311, 313, 331, 333, 335, 342, 347, 355, 367, 381, 385, 391, 395, 397, 406

Offset: 1

Labos Elemer

nonn

Cf. A034693, A034694.

A034849 a(n) = 1 + 2*A034784(n).

Original entry on oeis.org

7, 11, 17, 19, 23, 29, 31, 41, 43, 47, 53, 59, 67, 71, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 163, 167, 173, 179, 181, 191, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 281, 283, 293, 307, 311, 317, 331

Offset: 1

Labos Elemer

nonn

The terms are odd prime numbers.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A034693, A034694, A034784.

1 + 2 Position[#, 2] &@ Table[k = 1; While[! PrimeQ[k n + 1], k++]; k, {n, 165}] // Flatten (* Michael De Vlieger, Jul 31 2017 *)

lista(nn) = for (n=1, nn, if (isprime(p=2*n+1) && !isprime(n+1), print1(p, ", "));); \\ Michel Marcus, Aug 01 2017

Edited by N. J. A. Sloane, Oct 27 2012

A072063 Smallest prime of form prime(n)+k*n, k>0.

Original entry on oeis.org

3, 5, 11, 11, 31, 19, 31, 43, 41, 59, 53, 61, 67, 71, 107, 101, 127, 79, 181, 131, 157, 101, 313, 113, 197, 127, 157, 163, 167, 173, 251, 163, 269, 173, 359, 223, 379, 239, 401, 293, 1163, 223, 277, 281, 467, 337, 587, 271, 521, 379, 641, 499

Offset: 1

Reinhard Zumkeller, Jun 12 2002

nonn

According to Dirichlet's theorem primes of form prime(n)+k*n exist for all n, as gcd(n, prime(n))=1.

Nontrivial least prime == prime(n) (mod n).

			n=3, prime(3)=5: 5+1*3=8 is not prime, but 5+2*3=11, therefore a(3)=11 and A072064(3)=2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A034694, A000040, A061068, A072064.

Cf. A090471, A090472, A090473.

f:= proc(n) local p,k,k0;
  p:= ithprime(n);
  if n::odd then k0:= 2 else k0:= 1 fi;
  for k from k0 by k0 do
    if isprime(p+k*n) then return p+k*n fi
  od:
end proc:
f(1):= 3:
map(f, [$1..100]); # Robert Israel, Nov 27 2023

sp[n_]:=Module[{p=Prime[n],k=1},While[!PrimeQ[p+k*n],k++];p+k*n]; Array[ sp,60] (* Harvey P. Dale, Apr 19 2019 *)

a072063(n) = {my (p=prime(n), j); for (k=1, oo, if(isprime(j=p+k*n), return(j)))}; \\ Hugo Pfoertner, Nov 27 2023

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}.

A060940 Triangle in which n-th row gives the phi(n) terms appearing as initial primes in arithmetic progressions with difference n, with initial term equal to the smallest positive residue coprimes to n.

Original entry on oeis.org

2, 3, 7, 5, 5, 7, 11, 7, 13, 19, 7, 11, 29, 23, 17, 11, 19, 13, 17, 11, 13, 23, 19, 11, 13, 23, 43, 17, 11, 13, 17, 19, 23, 13, 47, 37, 71, 17, 29, 19, 31, 43, 13, 17, 19, 23, 53, 41, 29, 17, 31, 19, 59, 47, 61, 23, 37, 103, 29, 17, 19, 23, 53, 41, 31, 17, 19, 37, 23, 41, 43, 29

Offset: 1

Labos Elemer, May 07 2001

			For differences 1, 2, 3, 4, 5, 6, 7, .. the initial primes are 2; 3; 7, 5; 5, 7; 11, 7, 13, 19; 7, 11; 29, 23, 17, 11, 19, 13; ... etc. Suitable initial values (coprimes to difference) are in A038566. Position of end(start) of rows is given by values of A002088.
From _Seiichi Manyama_, Apr 02 2018: (Start)
   n | phi(n)|
  ---+-------+------------------------
   1 |   1   |  2;
   2 |   1   |  3;
   3 |   2   |  7,  5;
   4 |   2   |  5,  7;
   5 |   4   | 11,  7, 13, 19;
   6 |   2   |  7, 11;
   7 |   6   | 29, 23, 17, 11, 19, 13;
   8 |   4   | 17, 11, 13, 23;
   9 |   6   | 19, 11, 13, 23, 43, 17;
  10 |   4   | 11, 13, 17, 19;         (End)

Seiichi Manyama, Rows n = 1..200, flattened
Eric Weisstein's MathWorld, Dirichlet's theorem
Wikipedia, Dirichlet's theorem on arithmetic progressions

Cf. A000010 (phi), A002088, A038566, A034693, A034694, A002144, A007519, A088732, etc..

A061599 Primes p such that the greatest prime divisor of p-1 is 5.

Original entry on oeis.org

11, 31, 41, 61, 101, 151, 181, 241, 251, 271, 401, 541, 601, 641, 751, 811, 1201, 1601, 1621, 1801, 2161, 2251, 3001, 4001, 4051, 4801, 4861, 6481, 7681, 8101, 8641, 9001, 9601, 9721, 11251, 14401, 15361, 16001, 19441, 21601, 21871, 22501, 23041, 24001

Offset: 1

Labos Elemer, Jun 13 2001

nonn

Prime numbers n for which cos(2Pi/n) is an algebraic number of 5th degree. - Artur Jasinski, Dec 13 2006

The least significant digit of each term is one. - Harvey P. Dale, Jul 07 2024

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

The 3rd in a family of sequences after A019434(=Fermat-primes) and A058383.
Cf. A019434, A058383, A023503, A034694, A006530, A006093, A035095, A000040.
Cf. A004729, A058383, A125867-A125875, A024899.

Do[If[Take[FactorInteger[EulerPhi[2n + 1]][[ -1]],1] == {5} && PrimeQ[2n + 1], Print[2n + 1]], {n, 1, 10000}] (* Artur Jasinski, Dec 13 2006 *)
Select[Prime[Range[3000]],Max[FactorInteger[#-1][[;;,1]]]==5&] (* Harvey P. Dale, Jul 07 2024 *)

{ default(primelimit, 167772161); n=0; forprime (p=3, 167772161, f=factor(p - 1)~; if (f[1, length(f)]==5, write("b061599.txt", n++, " ", p)) ) } \\ Harry J. Smith, Jul 25 2009

list(lim)=my(v=List(), s, t); lim\=1; lim--; for(i=1, logint(lim\2, 5), t=2*5^i; for(j=0, logint(lim\t, 3), s=t*3^j; while(s<=lim, if(isprime(s+1), listput(v, s+1)); s<<=1))); Set(v) \\ Charles R Greathouse IV, Oct 29 2018

Primes of the form 2^a*3^b*5^c + 1 with a and c > 0.

A077319 Average of terms in n-th row of A077316.

Original entry on oeis.org

2, 4, 13, 16, 43, 25, 113, 95, 131, 97, 329, 139, 479, 240, 373, 384, 813, 302, 1315, 514, 815, 675, 1555, 775, 1697, 1015, 1607, 1235, 3189, 806, 3739, 1795, 2407, 1800, 3303, 1537, 5349, 2790, 3787, 2652, 6705, 1869, 7255, 3645, 4663, 3504, 8411, 3097

Offset: 1

Amarnath Murthy, Nov 04 2002

nonn

Cf. A034694, A077316, A077317, A077318.

a(n) = A077318(n)/n.

Edited, corrected and extended by Franklin T. Adams-Watters, Aug 29 2006

A279756 Smallest prime p >= prime(n) such that p == 2 (mod prime(n)).

Original entry on oeis.org

2, 5, 7, 23, 13, 41, 19, 59, 71, 31, 157, 113, 43, 131, 331, 373, 61, 307, 337, 73, 367, 239, 251, 269, 293, 103, 311, 109, 547, 1019, 383, 919, 139, 419, 151, 757, 787, 491, 503, 521, 181, 907, 193, 967, 199, 599, 1901, 1117, 229, 2063, 701, 241, 3617, 1759, 773

Offset: 1

Thomas Ordowski and Altug Alkan, Dec 18 2016

nonn

Conjecture: a(n) < prime(n)^2 for every n.

If this conjecture is true, then no terms are repeated.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

A006512 is a subsequence.

Cf. A000040, A034694.

a:= proc(n) local q, p; p:= ithprime(n); q:= p;
      do if irem(q-2, p)=0 then break fi;
         q:= nextprime(q);
      od; q
    end:
seq(a(n), n=1..55);  # Alois P. Heinz, May 03 2021

a[n_] := Module[{p = Prime[n], q}, q = p; While[True, If[Mod[q-2, p] == 0, Break[], q = NextPrime[q]]]; q];
Table[a[n], {n, 1, 55}] (* Jean-François Alcover, Jun 13 2025, after Alois P. Heinz *)

a(n) = {p = prime(n); q = p; while (Mod(q, p) != 2, q = nextprime(q+1)); q;} \\ Michel Marcus, Dec 18 2016

from itertools import dropwhile, count
from sympy import isprime, prime
def A279756(n): return next(dropwhile(lambda x:not isprime(x),count(2 if (p:=prime(n))==2 else p+2,p))) # Chai Wah Wu, Jan 04 2024

A035091 Smallest prime == 1 mod (n^2).

Original entry on oeis.org

2, 5, 19, 17, 101, 37, 197, 193, 163, 101, 727, 433, 677, 197, 1801, 257, 3469, 1297, 10831, 401, 883, 1453, 12697, 577, 11251, 677, 1459, 3137, 10093, 1801, 15377, 12289, 2179, 3469, 7351, 1297, 5477, 18773, 9127, 1601, 16811, 3529, 22189, 11617

Offset: 1

Labos Elemer

nonn

Smallest prime of form (n^2)*k+1, i.e., an arithmetic progression with n^2 differences; k is the subscript of the progressions.

			a(5) = 101 because in 5^2k + 1 = 25k + 1 progression k=4 generates the smallest prime (this is 101) and 26, 51, and 76 are composite.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..500 from Robert Price)
Index entries for sequences related to primes in arithmetic progressions.

Analogous case is A034694. Special case is A002496.

Cf. A070844 to A070858, A061092, A035095.

With[{prs=Prime[Range[2500]]},Flatten[Table[Select[prs,Mod[#-1,n^2]==0&,1],{n,50}]]] (* Harvey P. Dale, Sep 22 2021 *)

a(n) = if(n == 1, 2, my(s = n^2); forprime(p = 1, , if(p % s == 1, return(p)))); \\ Amiram Eldar, Mar 16 2025

cp's OEIS Frontend

A034780 Numbers k such that A034693(k) = 4.

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A034783 Numbers k such that A034693(k) = 6.

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A034849 a(n) = 1 + 2*A034784(n).

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A072063 Smallest prime of form prime(n)+k*n, k>0.

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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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A060940 Triangle in which n-th row gives the phi(n) terms appearing as initial primes in arithmetic progressions with difference n, with initial term equal to the smallest positive residue coprimes to n.

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A061599 Primes p such that the greatest prime divisor of p-1 is 5.

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Mathematica

PARI

PARI

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A077319 Average of terms in n-th row of A077316.

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Extensions

A279756 Smallest prime p >= prime(n) such that p == 2 (mod prime(n)).

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