A070858 -id:A070858 - cp's OEIS frontend

A035095 Smallest prime congruent to 1 (mod prime(n)).

3, 7, 11, 29, 23, 53, 103, 191, 47, 59, 311, 149, 83, 173, 283, 107, 709, 367, 269, 569, 293, 317, 167, 179, 389, 607, 619, 643, 1091, 227, 509, 263, 823, 557, 1193, 907, 1571, 653, 2339, 347, 359, 1087, 383, 773, 3547, 797, 2111, 2677, 5449, 2749, 467

Offset: 1

Labos Elemer

nonn

This is a version of the "least prime in special arithmetic progressions" problem.
Smallest numbers m such that largest prime factor of Phi(m) = prime(n), the n-th prime, also seems to be prime and identical to n-th term of A035095. See A068211, A068212, A065966: Min[x : A068211(x)=prime(n)] = A035095(n); e.g., Phi(a(7)) = Phi(103) = 2*3*17, of which 17 = p(7) is the largest prime factor, arising first here.
It appears that A035095, A066674, A125878 are probably all the same, but see the comments in A066674. - N. J. A. Sloane, Jan 05 2013
Minimum of the smallest prime factors of F(n,i) = (i^prime(n)-1)/(i-1), when i runs through all integers in [2, prime(n)]. Every prime factor of F(n,i) is congruent to 1 modulo prime(n). - Vladimir Shevelev, Nov 26 2014
Conjecture: a(n) is the smallest prime p such that gpf(p-1) = prime(n). See A023503. - Thomas Ordowski, Aug 06 2017
For n>1, a(n) is the smallest prime congruent to 1 mod (2*prime(n)). - Chai Wah Wu, Apr 28 2025

			a(8) = 191 because in the prime(8)k+1 = 19k+1 sequence, 191 is the smallest prime.

E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Bd 1 (reprinted Chelsea 1953).
E. C. Titchmarsh, A divisor problem, Renc. Circ. Math. Palermo, 54 (1930) pp. 414-429.
P. Turan, Über Primzahlen der arithmetischen Progression, Acta Sci. Math. (Szeged), 8 (1936/37) pp. 226-235.

T. D. Noe, Table of n, a(n) for n = 1..10000
P. Erdős, On some application of Brun's method, Acta Sci. Math (Szeged), v. 13, 1949, pp. 57-63.
A. Granville and C. Pomerance, On the least prime in certain arithmetic progressions
A. Granville and C. Pomerance, On the least prime in certain arithmetic progressions J. Lond Math Soc s2-41 (2) (1990), pp. 193-200.
D. R. Heath-Brown, almost-primes in arithmetic progressions in short intervals, Math Proc Cambr. Phil Soc v 83 (1978), pp. 357-375.
D. R. Heath-Brown, Siegel zeros and the least prime in arithmetic progression, Quart. J. of Math 41 (49) (1990), pp. 405-418.
H.-J. Kanold, Uber Primzahlen in arithmetischen Folgen, Math. Ann. v 156 (1964) pp. 393-395.
U. V. Linnik, On the least prime in an arithmetic progression. I. The basic theorem, Rec. Math (N.S.) v 15 (57) (1944), pp 139-178. MR0012111
C. Pomerance, A note on the least prime in an arithmetic progression J. Number Theory 12 (2) (1980), pp. 218-223.
K. Prachar, Uber die kleinste Primzahl in einer arithmetischen Reihe, J Reine Angew Math. 206 (1961) pp. 3-4.
A. Schinzel, Remark on the paper of K. Prachar Uber die kleinste.., J. Reine Angew Math. v 210 (1962) pp. 122-122.
S. S. Wagstaff, Jr., The irregular primes to 125000, Math. Comp., 32 (1978) pp. 583-591.
S. S. Wagstaff, Jr, Greatest of the Least Primes in Arithmetic Progressions Having a Given Modulus, Math. Comp., 33 (147) (1979) pp. 1073-1080.
Index entries for sequences related to primes in arithmetic progressions

Cf. A034694, A032448, A006530, A006093, A035096, A000040, A019434, A058383.
Cf. A068211, A068212, A065966, A000010, A070844-A070858, A061092.
Cf. A000040.

a[n_] := Block[{p = Prime[n]}, r = 1 + p; While[ !PrimeQ[r], r += p]; r]; Array[a, 51] (* Jean-François Alcover, Sep 20 2011, after PARI *)
a[n_]:=If[n<2,3,Block[{p=Prime[n]},r=1+2*p;While[!PrimeQ[r],r+=2*p]];r];Array[a,51] (* Zak Seidov, Dec 14 2013 *)

a(n)=local(p,r);p=prime(n);r=1;while(!isprime(r),r+=p);r

{my(N=66); forprime(p=2, , forprime(q=p+1,10^10, if((q-1)%p==0, print1(q,", "); N-=1; break)); if(N==0,break)); } \\ Joerg Arndt, May 27 2016

from itertools import count
from sympy import prime, isprime
def A035095(n): return 3 if n==1 else next(filter(isprime,count((p:=prime(n)<<1)+1,p))) # Chai Wah Wu, Apr 28 2025

According to a long-standing conjecture (see the 1979 Wagstaff reference), a(n) <= prime(n)^2 + 1. This would be sufficient to imply that a(n) is the smallest prime such that greatest prime divisor of a(n)-1 is prime(n), the n-th prime: A006530(a(n)-1) = A000040(n). This in turn would be sufficient to imply that no value occurs twice in this sequence. - Franklin T. Adams-Watters, Jun 18 2010

a(n) = 1 + A035096(n)*A000040(n). - Zak Seidov, Dec 27 2013

Edited by Franklin T. Adams-Watters, Jun 18 2010
Minor edits by N. J. A. Sloane, Jun 27 2010
Edited by N. J. A. Sloane, Jan 05 2013

A049536 Primes of the form lcm(1, ..., n) + 1 = A003418(n) + 1.

Original entry on oeis.org

2, 3, 7, 13, 61, 421, 2521, 232792561, 26771144401, 72201776446801, 442720643463713815201, 718766754945489455304472257065075294401, 6676878045498705789701874602220118271269436344024536001

Offset: 1

Labos Elemer

nonn

			Lcm(9) + 1 = lcm(10) + 1 = 2521, a prime.

Harvey P. Dale, Table of n, a(n) for n = 1..16

Subsequence of A070858.

Cf. A003418, A049537, A057824.

Select[Table[LCM@@Range[n]+1,{n,150}],PrimeQ]//Union (* Harvey P. Dale, May 31 2017 *)

N=1; print1(2); for(n=1,1e3, if(isprimepower(n,&p), N*=p; if(isprime(N+1), print1(", "N+1)))) \\ Charles R Greathouse IV, Nov 18 2015

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

A083685 a(n) is the smallest prime of the form k*lcm(1..n) - 1.

Original entry on oeis.org

2, 3, 5, 11, 59, 59, 419, 839, 5039, 5039, 55439, 55439, 1081079, 1081079, 1081079, 1441439, 24504479, 24504479, 232792559, 232792559, 232792559, 232792559, 5354228879, 5354228879, 53542288799, 53542288799, 1044074631599

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 15 2003

nonn

			a(9) = 5039: lcm(1..9) = 2520; 2519 is not a prime but 2*2520 - 1 = 5039 is a prime.

Harvey P. Dale, Table of n, a(n) for n = 1..500

Cf. A070858.

spf[n_]:=Module[{c=LCM@@Range[n],k=1},While[!PrimeQ[k*c-1],k++];k*c-1]; Array[spf,30] (* Harvey P. Dale, Jan 15 2022 *)

a(n) = {my(lcmn = lcm(vector(n, k, k)), k = 1); while(!isprime(p = k*lcmn-1), k++); p;} \\ Michel Marcus, Mar 15 2018

Corrected and extended by Vladeta Jovovic, Jun 16 2003

A272857 Least k>1 such that the Euler totient function of powers k^e, 1 <= e <= n, are divisible by the number their divisors, d(k^e).

Original entry on oeis.org

3, 3, 13, 61, 61, 421, 2521, 2521, 2521, 55441, 55441, 4324321, 4324321, 4324321, 4324321, 85765681, 85765681, 232792561, 232792561, 232792561, 232792561

Offset: 1

Paolo P. Lava, May 12 2016

a(22) <= 10708457761. - Amiram Eldar, May 27 2024

			phi(3) / d(3) = 2 / 2 = 1, phi(3^2) / d(3^2) = 6 / 3 = 2 but phi(3^3) / d(3^3) = 18 / 4 = 9 / 2;
phi(13) / d(13) = 12 / 2 = 6, phi(13^2) / d(13^2) = 156 / 3 = 52, phi(13^3) / d(13^3) = 2028 / 4 = 507 but phi(13^4) / d(13^4) = 26364 / 5.

Cf. A000005, A000010, A070858, A272980, A272981.

with(numtheory): P:= proc(q) local a, j, k, ok, p; global n; a:=2;
for k from 1 to q do for n from a to q do ok:=1;
for j from 1 to k do if not type(phi(n^j)/tau(n^j), integer) then ok:=0; break; fi; od;
if ok=1 then a:=n; print(n); break; fi; od; od; end: P(10^9);

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A035095 Smallest prime congruent to 1 (mod prime(n)).

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A049536 Primes of the form lcm(1, ..., n) + 1 = A003418(n) + 1.

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

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PARI

A083685 a(n) is the smallest prime of the form k*lcm(1..n) - 1.

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A272857 Least k>1 such that the Euler totient function of powers k^e, 1 <= e <= n, are divisible by the number their divisors, d(k^e).

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