A034694 -id:A034694 - cp's OEIS frontend

A071899 a(0)=1 and for n>0: a(n) = least prime>a(n-1) of form 1+k*n.

1, 2, 3, 7, 13, 31, 37, 43, 73, 109, 131, 199, 229, 313, 337, 421, 433, 443, 487, 571, 601, 631, 661, 691, 769, 1051, 1093, 1297, 1373, 1451, 1471, 1489, 1601, 1783, 1871, 2311, 2341, 2591, 2699, 2731, 2801, 2953, 3067, 3527, 3697, 4051

Offset: 0

Reinhard Zumkeller, Jun 12 2002

			n=5: in the sequence 1+k*5: 6,11,16,21,26,31,36,41,... the first prime >13=a(4) is 31, therefore a(5)=31.

Cf. A034694, A000040.

a[0] = 1; a[1] = 2; a[n_] := a[n] = (p = NextPrime[a[n-1]]; While[True, If[Mod[p, n] == 1, Break[], p = NextPrime[p]]]; p); Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Oct 10 2011 *)
lp[{n_,p_}]:=Module[{np=NextPrime[p]},While[!IntegerQ[(np-1)/(n+1)],np= NextPrime[np]]; {n+1,np}]; Transpose[NestList[lp,{0,1},50]][[2]] (* Harvey P. Dale, May 12 2015 *)

A085641 Smallest prime == 1 (mod pq...*k) where p, q, ..., k are all the distinct prime divisors of n. Or, smallest prime == 1 (mod the largest squarefree divisor of n).

Original entry on oeis.org

2, 3, 7, 3, 11, 7, 29, 3, 7, 11, 23, 7, 53, 29, 31, 3, 103, 7, 191, 11, 43, 23, 47, 7, 11, 53, 7, 29, 59, 31, 311, 3, 67, 103, 71, 7, 149, 191, 79, 11, 83, 43, 173, 23, 31, 47, 283, 7, 29, 11, 103, 53, 107, 7, 331, 29, 229, 59, 709, 31, 367, 311, 43, 3, 131, 67, 269, 103, 139

Offset: 1

Amarnath Murthy, Jason Earls, Jul 11 2003

All the numbers having the same set of prime divisors are mapped to the same prime.

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A007947, A034694.

a(n)={my(m=vecprod(factor(n)[,1]), p=1); while(!isprime(p), p+=m); p} \\ Andrew Howroyd, Dec 10 2024

a(n) = A034694(A007947(n)).

Offset corrected by Andrew Howroyd, Dec 10 2024

A086507 If n is even, a(n) = smallest prime == 1 (mod n), If n is odd, a(n) = smallest prime == -1 (mod n).

Original entry on oeis.org

2, 3, 2, 5, 19, 7, 13, 17, 17, 11, 43, 13, 103, 29, 29, 17, 67, 19, 37, 41, 41, 23, 137, 73, 149, 53, 53, 29, 173, 31, 61, 97, 131, 103, 139, 37, 73, 191, 233, 41, 163, 43, 257, 89, 89, 47, 281, 97, 97, 101, 101, 53, 211, 109, 109, 113, 113, 59, 353, 61, 487, 311, 251

Offset: 1

Amarnath Murthy, Jul 29 2003

nonn

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

Cf. A034694, A038700, A086508.

f:= proc(n) local x;
if n::even then x:= 1 else x:= -1; fi;
do
  x:= x+n;
  if isprime(x) then return x fi
od
end proc:
map(f, [$1..100]); # Robert Israel, Dec 09 2020

More terms from David Wasserman, Mar 09 2005

A087952 Smallest prime == 1 (mod n) and > n^2.

Original entry on oeis.org

2, 5, 13, 17, 31, 37, 71, 73, 109, 101, 199, 157, 313, 197, 241, 257, 307, 379, 419, 401, 463, 617, 599, 577, 701, 677, 757, 953, 929, 991, 1117, 1153, 1123, 1259, 1471, 1297, 1481, 1483, 1873, 1601, 1723, 1933, 1979, 2069, 2161, 2347, 2351, 2593, 2549, 2551

Offset: 1

Ray Chandler, Sep 16 2003

nonn

Primes arising in A087554.

Since A014085(n) ~ n/log(n) one may conjecture that a(n) < 2*n^2 for all n > 1. Numerically we find a(n) = n^2*(1 + O(1/sqrt(n))). - M. F. Hasler, Feb 27 2020

			For n=1, a(1) = 2, because 2 == 1 mod 1 and 2 > 1^2.
For n=2, a(2) = 5, because 5 == 1 mod 2 and 5 > 2^2.

M. F. Hasler, Table of n, a(n) for n = 1..10000, Feb 27 2020

Cf. A034693, A034694, A087554.

Cf. A014085 (number of primes between n^2 and (n+1)^2).

spr[n_]:=Module[{p=NextPrime[n^2]},While[Mod[p,n]!=1,p=NextPrime[p]];p]; Join[ {2},Array[spr,50,2]] (* Harvey P. Dale, Jun 21 2021 *)

apply( {A087952(n)=forprime(p=n^2+1,,(p-1)%n||return(p))}, [1..66]) \\ M. F. Hasler, Feb 27 2020

Examples added by N. J. A. Sloane, Jun 21 2021

A116605 Smallest prime p such that p == 1 (mod prime(n)) and not p == 1 (mod k) for 2 < k < prime(n).

Original entry on oeis.org

3, 7, 11, 239, 23, 443, 647, 1103, 47, 59, 2543, 3923, 83, 9203, 6299, 107, 7907, 8663, 11927, 14627, 12119, 15959, 167, 179, 20759, 20807, 23279, 23327, 28559, 227, 37847, 263, 43019, 54767, 53939, 54059, 54323, 54443, 66467, 347, 359, 69143, 383

Offset: 1

Klaus Brockhaus, Feb 19 2006

nonn

a(n) > 2*prime(n) for n > 1.

a(n) = 2*prime(n)+1 if prime(n) is in A005384. Otherwise, a(n) > 2*prime(n)^2+1 for n > 1. - Robert Israel, Mar 29 2017

			a(1) = 3 since prime(1) = 2 and 3 == 1 (mod 2).
a(4) = 239 since prime(4) = 7, 239 == 1 (mod 7) and for each of the primes q smaller than 239 with q == 1 (mod 7) there is a k (2 < k < 7) such that q == 1 (mod k): 29 == 1 (mod 4), 43 == 1 (mod 6), 71 == 1 (mod 5), 113 == 1 (mod 4), 127 == 1 (mod 3), 197 == 1 (mod 4), 211 == 1 (mod 5), whereas 239 == 2 (mod 3), 3 (mod 4), 4 (mod 5), 5
(mod 6).

Robert Israel, Table of n, a(n) for n = 1..408

Cf. A034694, A116606.

V:= {seq(4*i+2,i=1..10^5)}: A[1]:= 3:
for n from 2 do
  pn:= ithprime(n);
  R:= select(t -> t mod pn = 0, V);
  found:= false;
  for r in R do
    if isprime(r+1) then
      found:= true;
      A[n]:= r+1;
      break
    fi
  od;
  if not found then break fi;
  V:= V minus R;
od:
seq(A[i],i=1..n-1); # Robert Israel, Mar 29 2017

A338929 a(n) is the smallest prime number p larger than A072668(n) such that p is equal to 1 (mod A072668(n)).

Original entry on oeis.org

7, 11, 29, 17, 19, 23, 53, 29, 31, 103, 191, 41, 43, 47, 73, 101, 53, 109, 59, 311, 97, 67, 103, 71, 149, 191, 79, 83, 173, 89, 181, 283, 97, 197, 101, 103, 107, 109, 331, 113, 229, 709, 367, 311, 127, 193, 131, 269, 137, 139, 569, 293, 149, 151, 229, 463

Offset: 1

Ahmad J. Masad, Nov 15 2020

nonn

In A002808(n)-base numeral system, a(n) is the smallest prime number for which the digital root is 1.
Conjecture: As n approaches infinity, the probability that a prime number is a term in this sequence approaches 1.
Conjecture: There are infinitely many primes that are not terms in this sequence.
The sequence for all positive numbers (instead of A072668) is A034694. - Peter Munn, May 02 2023

			For n=20, A072668(20)=31, and 311 is the smallest prime number p larger than 31 such that p is equal to 1 (mod 31), so a(20)=311.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A000040, A002808, A034694, A072668, A339540.

Map[Block[{p = NextPrime[#]}, While[Mod[p, #] != 1, p = NextPrime[p]]; p] &, Select[Range[4, 78], CompositeQ] - 1] (* Michael De Vlieger, Dec 10 2020 *)

f(x) = {my(p=nextprime(x)); while ((p % x) != 1, p = nextprime(p+1)); p;}
lista(nn) = {my(list = List()); forcomposite(c=1, nn, listput(list, f(c-1));); Vec(list);} \\ Michel Marcus, Nov 17 2020

More terms from Michel Marcus, Nov 17 2020

A034846 a(n) = P(n,6) = 1+6K(n,6)=1+6A034783(n). P(n,6) are special primes of 6k+1. The relevant values of k are given by A034783.

Original entry on oeis.org

103, 283, 331, 367, 463, 547, 607, 619, 643, 709, 727, 739, 823, 859, 883, 907, 967, 1021, 1087, 1123, 1171, 1249, 1303, 1423, 1447, 1483, 1489, 1543, 1579, 1597, 1627, 1699, 1723, 1747, 1783

Offset: 1

Labos Elemer

nonn

Cf. A034693, A034694, A034783.

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

Original entry on oeis.org

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

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

Labos Elemer

nonn

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.

Cf. A034693, A034694, A034782.

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

Corrected (wrong term 769 removed) and extended by Michel Marcus, May 13 2018

A061638 Primes p such that the greatest prime divisor of p-1 is 7.

Original entry on oeis.org

29, 43, 71, 113, 127, 197, 211, 281, 337, 379, 421, 449, 491, 631, 673, 701, 757, 883, 1009, 1051, 1373, 1471, 2017, 2269, 2521, 2647, 2689, 2801, 3137, 3361, 3529, 4201, 4481, 5881, 6301, 7001, 7057, 7351, 7561, 7841, 8233, 8821, 10501, 10753, 12097

Offset: 1

Labos Elemer, Jun 13 2001

nonn

Prime numbers n for which cos(2*Pi/n) is an algebraic number of 7th degree. - Artur Jasinski, Dec 13 2006

			For n = {4, 8, 9, 12}, a(n)-1 = {70, 210, 280, 420} = 7*{10, 30, 40, 60}.

Harry J. Smith and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 500 terms from Smith)

The 4th in a family of sequences after A019434(=Fermat-primes), A058383, A061599.
Cf. A019434, A058383, A023503, A034694, A006530, A006093, A035095, A061599.
Cf. A004729, A058383, A125867-A125875, A024899.

Select[Prime[Range[2000]],FactorInteger[#-1][[-1,1]] ==7&]  (* Harvey P. Dale, Mar 12 2011 *)

default(primelimit, 108864001); n=0; forprime (p=3, 108864001, f=factor(p - 1)~; if (f[1, length(f)]==7, write("b061638.txt", n++, " ", p))) \\ Harry J. Smith, Jul 25 2009

list(lim)=my(v=List(), t, t5, t7); lim\=1; lim--; for(a=1, logint(lim\2, 7), t7=2*7^a; for(b=0, logint(lim\t7, 5), t5=5^b*t7; for(c=0, logint(lim\t5, 3), t=3^c*t5; while(t<=lim, if(isprime(t+1), listput(v, t+1)); t<<=1)))); Set(v) \\ Charles R Greathouse IV, Oct 29 2018

Primes of form 2^a*3^b*5^c*7^d + 1 with a and d > 1.

cp's OEIS Frontend

A071899 a(0)=1 and for n>0: a(n) = least prime>a(n-1) of form 1+k*n.

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A085641 Smallest prime == 1 (mod p*q*...*k) where p, q, ..., k are all the distinct prime divisors of n. Or, smallest prime == 1 (mod the largest squarefree divisor of n).

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A086507 If n is even, a(n) = smallest prime == 1 (mod n), If n is odd, a(n) = smallest prime == -1 (mod n).

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Maple

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A087952 Smallest prime == 1 (mod n) and > n^2.

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A116605 Smallest prime p such that p == 1 (mod prime(n)) and not p == 1 (mod k) for 2 < k < prime(n).

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A338929 a(n) is the smallest prime number p larger than A072668(n) such that p is equal to 1 (mod A072668(n)).

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A034846 a(n) = P(n,6) = 1+6*K(n,6)=1+6*A034783(n). P(n,6) are special primes of 6k+1. The relevant values of k are given by A034783.

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

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PARI

Extensions

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

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A085641 Smallest prime == 1 (mod pq...*k) where p, q, ..., k are all the distinct prime divisors of n. Or, smallest prime == 1 (mod the largest squarefree divisor of n).

A034846 a(n) = P(n,6) = 1+6K(n,6)=1+6A034783(n). P(n,6) are special primes of 6k+1. The relevant values of k are given by A034783.