A034694 -id:A034694 - cp's OEIS frontend

A083809 Let f(n) be the smallest prime == 1 mod n (cf. A034694). Sequence gives triangle T(j,k) = f^k(j) for 1 <= k <= j, read by rows.

2, 3, 7, 7, 29, 59, 5, 11, 23, 47, 11, 23, 47, 283, 1699, 7, 29, 59, 709, 2837, 22697, 29, 59, 709, 2837, 22697, 590123, 1180247, 17, 103, 619, 2477, 34679, 416149, 7490683, 29962733, 19, 191, 383, 4597, 27583, 330997, 9267917, 74143337, 1038006719

Offset: 1

Amarnath Murthy, May 08 2003

It has been proved in the reference that for every prime p there exists a prime of the form k*p+1. Conjecture: sequence is infinite, i.e., for every n there exists a prime of the form n*k+1 (cf. A034693).

Both follow directly from Dirichlet's theorem. [Charles R Greathouse IV, Feb 28 2012]

			The first few rows of the triangle are
2
3 7
7 29 59
5 11 23 47
11 23 47 283 1699
7 29 59 709 2837 22697

Amarnath Murthy, On the divisors of Smarandache Unary Sequence. Smarandache Notions Journal, Vol. 11, 2000.

Vincenzo Librandi, Table of n, a(n) for n = 1..591

The first column is given by A034694; the sequence of the last terms in the rows (main diagonal) is A083810. Row sums are in A160940.

Cf. A034693.

f:=function(n) m:=1; while not IsPrime(m*n+1) do m+:=1; end while; return m*n+1; end function; &cat[ [ k eq 1 select f(j) else f(Self(k-1)): k in [1..j] ]: j in [1..9] ]; // Klaus Brockhaus, May 30 2009

f[1]=2; f[n_] := f[n] = Block[{p=2}, While[Mod[p, n] != 1, p = NextPrime[p]]; p];
Flatten[Table[Rest @ NestList[f, j, j], {j, 9}]]
(* Jean-François Alcover, May 31 2011, improved by Robert G. Wilson v *)

f(n)=my(k=n+1);while(!isprime(k),k+=n);k
T(j,k)=for(i=1,k,j=f(j));j \\ Charles R Greathouse IV, Feb 28 2012

Edited, corrected and extended by Klaus Brockhaus, May 13 2003

A120857 Positions of records in A034694.

Original entry on oeis.org

1, 2, 3, 5, 7, 13, 17, 19, 31, 55, 59, 85, 109, 133, 159, 197, 227, 317, 361, 415, 457, 521, 799, 859, 1073, 1163, 1637, 1691, 1861, 1997, 2053, 2633, 3011, 3167, 3397, 3721, 3833, 5207, 7633, 9199, 11603, 13469, 13549, 14099, 16103, 17483, 17693, 18113

Offset: 1

Robert G. Wilson v, Jul 08 2006

nonn

Daniel Mondot, Table of n, a(n) for n = 1..89

Cf. A034694, A120856.

f[n_] := Block[{k = 1}, If[n == 1, 2, While[ Mod[Prime@k, n] != 1, k++ ]; Prime@k]]; lst = {}; a = 1; Do[b = f@n; If[b > a, a = b; AppendTo[lst, {n, a}]; Print@{n, a}], {n, 18200}]; (Transpose@lst)[[1]]

Alois P. Heinz (Dec 18 2023) comments that a(90)= 1126291, a(100) = 2545019, a(111) = 10084901. - N. J. A. Sloane, Jan 22 2024

A083810 a(n) = f^(n) (n), where f(n) is the smallest prime == 1 mod n (cf. A034694).

Original entry on oeis.org

2, 7, 59, 47, 1699, 22697, 1180247, 29962733, 1038006719, 411043979, 295951664941, 439109442193, 84309012901073, 1566310690709, 399766531626388768981, 71604415998366709, 180443128315884106709

Offset: 1

Amarnath Murthy, May 08 2003

nonn

Main diagonal of A083809.

Cf. A034694, A083809.

Edited, corrected and extended by Klaus Brockhaus, May 13 2003

A120856 Records in A034694.

Original entry on oeis.org

2, 3, 7, 11, 29, 53, 103, 191, 311, 331, 709, 1021, 1091, 1597, 3181, 3547, 5449, 8243, 10831, 11621, 13711, 16673, 17579, 18899, 32191, 37217, 62207, 71023, 74441, 87869, 94439, 94789, 96353, 114013, 115499, 171167, 229981, 437389, 457981

Offset: 1

Robert G. Wilson v, Jul 08 2006

nonn

Cf. A034694, A120857.

f[n_] := Block[{k = 1}, If[n == 1, 2, While[ Mod[Prime@k, n] != 1, k++ ]; Prime@k]]; lst = {}; a = 1; Do[b = f@n; If[b > a, a = b; AppendTo[lst, {n, a}]; Print@{n, a}], {n, 18200}]; (Transpose@lst)[[2]]

A093416 Index of first occurrence of prime(n) in A034694.

Original entry on oeis.org

1, 2, 4, 3, 5, 12, 8, 9, 11, 7, 15, 36, 20, 21, 23, 13, 29, 60, 33, 35, 24, 39, 41, 44, 32, 25, 17, 53, 27, 56, 63, 65, 68, 69, 37, 75, 156, 81, 83, 43, 89, 45, 19, 64, 49, 99, 105, 111, 113, 57, 116, 119, 80, 125, 128, 131, 67, 135, 92, 140, 47, 73, 153, 31, 104, 79, 55, 84

Offset: 1

Klaus Brockhaus, Feb 13 2006

nonn

a(n) = smallest k such that A034694(k) = prime(n).

			A034694(7) = 29 = prime(10) and A034694(k) <> 29 for k < 7, hence a(10) = 7.
A034694(8) = 17 = prime(7) and A034694(k) <> 17 for k < 8, hence a(7) = 8.

Cf. A034694.

A006093 a(n) = prime(n) - 1.

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, 156, 162, 166, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 250, 256, 262, 268, 270

Offset: 1

N. J. A. Sloane

These are also the numbers that cannot be written as i*j + i + j (i,j >= 1). - Rainer Rosenthal, Jun 24 2001; Henry Bottomley, Jul 06 2002
The values of k for which Sum_{j=0..n} (-1)^j*binomial(k, j)*binomial(k-1-j, n-j)/(j+1) produces an integer for all n such that n < k. Setting k=10 yields [0, 1, 4, 11, 19, 23, 19, 11, 4, 1, 0] for n = [-1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], so 10 is in the sequence. Setting k=3 yields [0, 1, 1/2, 1/2] for n = [-1, 0, 1, 2], so 3 is not in the sequence. - Dug Eichelberger (dug(AT)mit.edu), May 14 2001
n such that x^n + x^(n-1) + x^(n-2) + ... + x + 1 is irreducible. - Robert G. Wilson v, Jun 22 2002
Records for Euler totient function phi.
Together with 0, n such that (n+1) divides (n!+1). - Benoit Cloitre, Aug 20 2002; corrected by Charles R Greathouse IV, Apr 20 2010
n such that phi(n^2) = phi(n^2 + n). - Jon Perry, Feb 19 2004
Numbers having only the trivial perfect partition consisting of a(n) 1's. - Lekraj Beedassy, Jul 23 2006
Numbers n such that the sequence {binomial coefficient C(k,n), k >= n } contains exactly one prime. - Artur Jasinski, Dec 02 2007
Record values of A143201: a(n) = A143201(A001747(n+1)) for n > 1. - Reinhard Zumkeller, Aug 12 2008
From Reinhard Zumkeller, Jul 10 2009: (Start)
The first N terms can be generated by the following sieving process:
  start with {1, 2, 3, 4, ..., N - 1, N};
  for i := 1 until SQRT(N) do
  (if (i is not striked out) then
  (for j := 2 * i + 1 step i + 1 until N do
  (strike j from the list)));
  remaining numbers = {a(n): a(n) <= N}. (End)
a(n) = partial sums of A075526(n-1) = Sum_{1..n} A075526(n-1) = Sum_{1..n} (A008578(n+1) - A008578(n)) = Sum_{1..n} (A158611(n+2) - A158611(n+1)) for n >= 1. - Jaroslav Krizek, Aug 04 2009
A006093 U A072668 = A000027. - Juri-Stepan Gerasimov, Oct 22 2009
A171400(a(n)) = 1 for n <> 2: subsequence of A171401, except for a(2) = 2. - Reinhard Zumkeller, Dec 08 2009
Numerator of (1 - 1/prime(n)). - Juri-Stepan Gerasimov, Jun 05 2010
Numbers n such that A002322(n+1) = n. This statement is stronger than repeating the property of the entries in A002322, because it also says in reciprocity that this sequence here contains no numbers beyond the Carmichael numbers with that property. - Michel Lagneau, Dec 12 2010
a(n) = A192134(A095874(A000040(n))); subsequence of A192133. - Reinhard Zumkeller, Jun 26 2011
prime(a(n)) + prime(k) < prime(a(k) + k) for at least one k <= a(n): A212210(a(n),k) < 0. - Reinhard Zumkeller, May 05 2012
Except for the first term, numbers n such that the sum of first n natural numbers does not divide the product of first n natural numbers; that is, n*(n + 1)/2 does not divide n!. - Jayanta Basu, Apr 24 2013
BigOmega(a(n)) equals BigOmega(a(n)*(a(n) + 1)/2), where BigOmega = A001222. Rationale: BigOmega of the product on the right hand side factorizes as BigOmega(a/2) + Bigomega(a+1) = BigOmega(a/2) + 1 because a/2 and a + 1 are coprime, because BigOmega is additive, and because a + 1 is prime. Furthermore Bigomega(a/2) = Bigomega(a) - 1 because essentially all 'a' are even. - Irina Gerasimova, Jun 06 2013
Record values of A060681. - Omar E. Pol, Oct 26 2013
Deficiency of n-th prime. - Omar E. Pol, Jan 30 2014
Conjecture: All the sums Sum_{k=s..t} 1/a(k) with 1 <= s <= t are pairwise distinct. In general, for any integers d >= -1 and m > 0, if Sum_{k=i..j} 1/(prime(k)+d)^m = Sum_{k=s..t} 1/(prime(k)+d)^m with 0 < i <= j and 0 < s <= t then we must have (i,j) = (s,t), unless d = m = 1 and {(i,j),(s,t)} = {(4,4),(8,10)} or {(4,7),(5,10)}. (Note that 1/(prime(8)+1)+1/(prime(9)+1)+1/(prime(10)+1) = 1/(prime(4)+1) and Sum_{k=5..10} 1/(prime(k)+1) = 1/(prime(4)+1) + Sum_{k=5..7} 1/(prime(k)+1).) - Zhi-Wei Sun, Sep 09 2015
Numbers n such that (prime(i)^n + n) is divisible by (n+1), for all i >= 1, except when prime(i) = n+1. - Richard R. Forberg, Aug 11 2016
a(n) is the period of Fubini numbers (A000670) over the n-th prime. - Federico Provvedi, Nov 28 2020

Archimedeans Problems Drive, Eureka, 40 (1979), 28.
Harvey Dubner, Generalized Fermat primes, J. Recreational Math., 18 (1985): 279-280.
M. Gardner, The Colossal Book of Mathematics, pp. 31, W. W. Norton & Co., NY, 2001.
M. Gardner, Mathematical Circus, pp. 251-2, Alfred A. Knopf, NY, 1979.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
Thomas F. Bloom, Unit fractions with shifted prime denominators, arXiv:2305.02689 [math.NT], 2023.
R. P. Boas & N. J. A. Sloane, Correspondence, 1974
Harvey Dubner, Generalized Fermat primes, J. Recreational Math. 18.4 (1985-1986), 279. (Annotated scanned copy)
Armel Mercier, Problem E 3065, American Mathematical Monthly, 1984, p. 649.
Armel Mercier, S. K. Rangarajan, J. C. Binz and Dan Marcus, Problem E 3065, American Mathematical Monthly, No. 4, 1987, pp. 378.
Poo-Sung Park, Additive uniqueness of PRIMES-1 for multiplicative functions, arXiv:1708.03037 [math.NT], 2017.
J. R. Rickard and J. J. Hitchcock, Problem Drive 4, Archimedeans Problems Drive, Eureka, 40 (1979), 28-29, 40. (Annotated scanned copy)
Index entries for sequences generated by sieves

a(n) = K(n, 1) and A034693(K(n, 1)) = 1 for all n. The subscript n refers to this sequence and K(n, 1) is the index in A034693. - Labos Elemer
Cf. A000040, A034694. Different from A075728.
Complement of A072668 (composite numbers minus 1), A072670(a(n))=0.
Essentially the same as A039915.
Cf. A084920, A006093, A050997, A008864, A060800, A131991, A131992, A131993.
Cf. A101301 (partial sums), A005867 (partial products).
Column 1 of the following arrays/triangles: A087738, A249741, A352707, A378979, A379010.
The last diagonal of A162619, and of A174996, the first diagonal in A131424.
Row lengths of irregular triangles A086145, A124223, A212157.

Filtered([1..280],IsPrime)-1; # Muniru A Asiru, Nov 25 2018

a006093 = (subtract 1) . a000040  -- Reinhard Zumkeller, Mar 06 2012

[NthPrime(n)-1: n in [1..100]]; // Vincenzo Librandi, Nov 17 2015

for n from 2 to 271 do if (n! mod n^2 = n*(n-1) and (n<>4) then print(n-1) fi od; # Gary Detlefs, Sep 10 2010
# alternative
A006093 := proc(n)
    ithprime(n)-1 ;
end proc:
seq(A006093(n),n=1..100) ; # R. J. Mathar, Feb 06 2019

Table[Prime[n] - 1, {n, 1, 30}] (* Vladimir Joseph Stephan Orlovsky, Apr 27 2008 *)
a[ n_] := If[ n < 1, 0, -1 + Prime @ n] (* Michael Somos, Jul 17 2011 *)
Prime[Range[60]] - 1 (* Alonso del Arte, Oct 26 2013 *)

isA006093(n) = isprime(n+1) \\ Michael B. Porter, Apr 09 2010

A006093(n) = prime(n)-1 \\ Michael B. Porter, Apr 09 2010

\\ Sieve as described in Rainer Rosenthal's comment.
m=270;s=vector(m);for(i=1,m,for(j=i,m,k=i*j+i+j;if(k<=m,s[k]=1)));for(k=1,m,if(s[k]==0,print1(k,", "))); \\ Hugo Pfoertner, May 14 2019

from sympy import prime
for n in range(1,100): print(prime(n)-1, end=', ') # Stefano Spezia, Nov 30 2018

a(n) = (p-1)! mod p where p is the n-th prime, by Wilson's theorem. - Jonathan Sondow, Jul 13 2010
a(n) = A000010(prime(n)) = A000010(A006005(n)). - Antti Karttunen, Dec 16 2012
a(n) = A005867(n+1)/A005867(n). - Eric Desbiaux, May 07 2013
a(n) = A000040(n) - 1. - Omar E. Pol, Oct 26 2013
a(n) = A033879(A000040(n)). - Omar E. Pol, Jan 30 2014

Correction for change of offset in A158611 and A008578 in Aug 2009 Jaroslav Krizek, Jan 27 2010
Obfuscating comments removed by Joerg Arndt, Mar 11 2010
Edited by Charles R Greathouse IV, Apr 20 2010

A007528 Primes of the form 6k-1.

Original entry on oeis.org

5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587

Offset: 1

N. J. A. Sloane

For values of k see A024898.
Also primes p such that p^q - 2 is not prime where q is an odd prime. These numbers cannot be prime because the binomial p^q = (6k-1)^q expands to 6h-1 some h. Then p^q-2 = 6h-1-2 is divisible by 3 thus not prime. - Cino Hilliard, Nov 12 2008
a(n) = A211890(3,n-1) for n <= 4. - Reinhard Zumkeller, Jul 13 2012
There exists a polygonal number P_s(3) = 3s - 3 = a(n) + 1. These are the only primes p with P_s(k) = p + 1, s >= 3, k >= 3, since P_s(k) - 1 is composite for k > 3. - Ralf Steiner, May 17 2018
From Bernard Schott, Feb 14 2019: (Start)
A theorem due to Andrzej Mąkowski: every integer greater than 161 is the sum of distinct primes of the form 6k-1. Examples: 162 = 5 + 11 + 17 + 23 + 47 + 59; 163 = 17 + 23 + 29 + 41 + 53. (See Sierpiński and David Wells.)
{2,3} Union A002476 Union {this sequence} = A000040.
Except for 2 and 3, all Sophie Germain primes are of the form 6k-1.
Except for 3, all the lesser of twin primes are also of the form 6k-1.
Dirichlet's theorem on arithmetic progressions states that this sequence is infinite. (End)
For all elements of this sequence p=6*k-1, there are no (x,y) positive integers such that k=6*x*y-x+y. - Pedro Caceres, Apr 06 2019

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
A. Mąkowski, Partitions into unequal primes, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 8 (1960), 125-126.
Wacław Sierpiński, Elementary Theory of Numbers, p. 144, Warsaw, 1964.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, Revised edition, 1997, p. 127.

T. D. Noe, Table of n, a(n) for n = 1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
F. S. Carey, On some cases of the Solutions of the Congruence z^p^(n-1)=1, mod p, Proceedings of the London Mathematical Society, Volume s1-33, Issue 1, November 1900, Pages 294-312.
Amelia Carolina Sparavigna, The Pentagonal Numbers and their Link to an Integer Sequence which contains the Primes of Form 6n-1, Politecnico di Torino (Italy, 2021).
Amelia Carolina Sparavigna, Binary operations inspired by generalized entropies applied to figurate numbers, Politecnico di Torino (Italy, 2021).
Wikipedia, Dirichlet's theorem on arithmetic progressions.

Intersection of A016969 and A000040.
Cf. A003627, A010051, A117047, A132231, A214360, A057145.
Prime sequences A# (k,r) of the form k*n+r with 0 <= r <= k-1 (i.e., primes == r (mod k), or primes p with p mod k = r) and gcd(r,k)=1: A000040 (1,0), A065091 (2,1), A002476 (3,1), A003627 (3,2), A002144 (4,1), A002145 (4,3), A030430 (5,1), A045380 (5,2), A030431 (5,3), A030433 (5,4), A002476 (6,1), this sequence (6,5), A140444 (7,1), A045392 (7,2), A045437 (7,3), A045471 (7,4), A045458 (7,5), A045473 (7,6), A007519 (8,1), A007520 (8,3), A007521 (8,5), A007522 (8,7), A061237 (9,1), A061238 (9,2), A061239 (9,4), A061240 (9,5), A061241 (9,7), A061242 (9,8), A030430 (10,1), A030431 (10,3), A030432 (10,7), A030433 (10,9), A141849 (11,1), A090187 (11,2), A141850 (11,3), A141851 (11,4), A141852 (11,5), A141853 (11,6), A141854 (11,7), A141855 (11,8), A141856 (11,9), A141857 (11,10), A068228 (12,1), A040117 (12,5), A068229 (12,7), A068231 (12,11).
Cf. A034694 (smallest prime == 1 (mod n)).
Cf. A038700 (smallest prime == n-1 (mod n)).
Cf. A038026 (largest possible value of smallest prime == r (mod n)).
Cf. A001359 (lesser of twin primes), A005384 (Sophie Germain primes).
Cf. A048265, A324076.

Filtered(List([1..100],n->6*n-1),IsPrime); # Muniru A Asiru, May 19 2018

a007528 n = a007528_list !! (n-1)
a007528_list = [x | k <- [0..], let x = 6 * k + 5, a010051' x == 1]
-- Reinhard Zumkeller, Jul 13 2012

select(isprime,[seq(6*n-1,n=1..100)]); # Muniru A Asiru, May 19 2018

Select[6 Range[100]-1,PrimeQ]  (* Harvey P. Dale, Feb 14 2011 *)

forprime(p=2, 1e3, if(p%6==5, print1(p, ", "))) \\ Charles R Greathouse IV, Jul 15 2011

forprimestep(p=5,1000,6, print1(p", ")) \\ Charles R Greathouse IV, Mar 03 2025

A003627 \ {2}. - R. J. Mathar, Oct 28 2008
Conjecture: Product_{n >= 1} ((a(n) - 1) / (a(n) + 1)) * ((A002476(n) + 1) / (A002476(n) - 1)) = 3/4. - Dimitris Valianatos, Feb 11 2020
From Vaclav Kotesovec, May 02 2020: (Start)
Product_{k>=1} (1 - 1/a(k)^2) = 9*A175646/Pi^2 = 1/1.060548293.... =4/(3*A333240).
Product_{k>=1} (1 + 1/a(k)^2) = A334482.
Product_{k>=1} (1 - 1/a(k)^3) = A334480.
Product_{k>=1} (1 + 1/a(k)^3) = A334479. (End)
Legendre symbol (-3, a(n)) = -1 and (-3, A002476(n)) = +1, for n >= 1. For prime 3 one sets (-3, 3) = 0. - Wolfdieter Lang, Mar 03 2021

A034693 Smallest k such that k*n+1 is prime.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 2, 1, 4, 2, 2, 1, 6, 1, 10, 2, 2, 1, 2, 3, 4, 2, 4, 1, 2, 1, 10, 3, 2, 3, 2, 1, 4, 5, 2, 1, 2, 1, 4, 2, 4, 1, 6, 2, 4, 2, 2, 1, 2, 2, 6, 2, 4, 1, 12, 1, 6, 5, 2, 3, 2, 1, 4, 2, 2, 1, 8, 1, 4, 2, 2, 3, 6, 1, 4, 3, 2, 1, 2, 4, 12, 2, 4, 1, 2, 2, 6, 3, 4, 3, 2, 1, 4, 2

Offset: 1

Labos Elemer

Conjecture: for every n > 1 there exists a number k < n such that n*k + 1 is a prime. - Amarnath Murthy, Apr 17 2001
A stronger conjecture: for every n there exists a number k < 1 + n^(.75) such that n*k + 1 is a prime. I have verified this up to n = 10^6. Also, the expression 1 + n^(.74) does not work as an upper bound (counterexample: n = 19). - Joseph L. Pe, Jul 16 2002
Stronger version of the conjecture verified up to 10^9. - Mauro Fiorentini, Jul 23 2023
It is known that, for almost all n, a(n) <= n^2. From Heath-Brown's result (1992) obtained with help of the GRH, it follows that a(n) <= (phi(n)*log(n))^2. - Vladimir Shevelev, Apr 30 2012
Conjecture: a(n) = O(log(n)*log(log(n))). - Thomas Ordowski, Oct 17 2014
I conjecture the opposite: a(n) / (log n log log n) is unbounded. See A194945 for records in this sequence. - Charles R Greathouse IV, Mar 21 2016

			If n=7, the smallest prime in the sequence 8, 15, 22, 29, ... is 29, so a(7)=4.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, section 2.12, pp. 127-130.
P. Ribenboim, (1989), The Book of Prime Number Records. Chapter 4, Section IV.B.: The Smallest Prime In Arithmetic Progressions, pp. 217-223.

T. D. Noe, Table of n, a(n) for n = 1..10000
Gunther Cornelissen, David Hokken, and Berend Ringeling, The asymptotic Mahler measure of Gaussian periods, arXiv:2507.09303 [math.NT], 2025. See p. 1.
Steven R. Finch, Linnik's Constant
D. Graham, On Linnik's Constant, Acta Arithm., 39, 1981, pp. 163-179.
D. R. Heath-Brown, Zero-free regions for Dirichlet L-functions, and the least prime in an arithmetic progression, Proc. London Math. Soc. 64(3) (1992), pp. 265-338.
Pengcheng Niu and Junli Zhang, On Two Conjectures of A. Murthy, ResearchGate (2024).
Ivan Niven and Barry Powell, Primes in Certain Arithmetic Progressions, Amer. Math. Monthly, 83, 1976, pp. 467-489.
Wikipedia, Dirichlet's theorem on arithmetic progressions

Cf. A010051, A034694, A053989, A071558, A085420, A103689, A194944 (records), A194945 (positions of records), A200996.

a034693 n = head [k | k <- [1..], a010051 (k * n + 1) == 1]
-- Reinhard Zumkeller, Feb 14 2013

A034693 := proc(n)
    for k from 1 do
        if isprime(k*n+1) then
            return k;
        end if;
    end do:
end proc: # R. J. Mathar, Jul 26 2015

a[n_]:=(k=0; While[!PrimeQ[++k*n + 1]]; k); Table[a[n], {n,100}] (* Jean-François Alcover, Jul 19 2011 *)

a(n)=if(n<0,0,s=1; while(isprime(s*n+1)==0,s++); s)

from sympy import isprime
def a(n):
    k = 1
    while not isprime(k*n+1): k += 1
    return k
print([a(n) for n in range(1, 99)]) # Michael S. Branicky, May 05 2022

It seems that Sum_{k=1..n} a(k) is asymptotic to (zeta(2)-1)*n*log(n) where zeta(2)-1 = Pi^2/6-1 = 0.6449... . - Benoit Cloitre, Aug 11 2002

a(n) = (A034694(n)-1) / n. - Joerg Arndt, Oct 18 2020

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

Original entry on oeis.org

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

cp's OEIS Frontend

A064673 Where the least prime p such that n = (p-1)/(q-1) and p > q is not the least prime == 1 (mod n) (A034694).

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A083809 Let f(n) be the smallest prime == 1 mod n (cf. A034694). Sequence gives triangle T(j,k) = f^k(j) for 1 <= k <= j, read by rows.

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A120857 Positions of records in A034694.

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A083810 a(n) = f^(n) (n), where f(n) is the smallest prime == 1 mod n (cf. A034694).

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A120856 Records in A034694.

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A093416 Index of first occurrence of prime(n) in A034694.

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A006093 a(n) = prime(n) - 1.

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A007528 Primes of the form 6k-1.

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