A035095 -id:A035095 - cp's OEIS frontend

A336280 Number of consecutive primes of the form k*prime(n) + 1, starting with the least such prime A035095(n), that divides the Wendt determinant A048954(prime(n)).

1, 1, 2, 1, 4, 3, 2, 3, 2, 3, 1, 5, 6, 2, 6, 3, 3, 3, 1, 6, 3, 5, 5, 7, 5, 5, 6, 7, 4, 7, 5, 10, 10, 4, 4, 6, 10, 3, 4, 12, 11, 5, 7, 8, 7, 8, 11, 4, 4, 4, 14, 7, 11, 7, 13, 11, 13, 7, 18, 18, 6, 7, 17, 12, 9, 7, 9, 14, 12, 9, 16, 14, 11, 13, 10

Offset: 1

Frank M Jackson and Michael B Rees, Jul 15 2020

nonn

Michael B Rees has conjectured that:
1. for every prime p, the Wendt determinant Wendt(p) has all its prime factors that are greater than p of the form k*p + 1.
2. for every prime p = prime(n) and its corresponding Wendt determinant W(p) there exists a finite number of m consecutive primes (p_1,p_2,..,p_m) of the form  k*p + 1 that will divide Wendt(p) where p_1 is always the least prime of the form k*p + 1.
This sequence gives the value m for each p = prime(n).

			a(6) = 3 gives p = prime(6) = 13 and W(13) = 3^6*53^2*79^2*131^2*521^2*8191. The sequence of primes of the form q = k*13 + 1, starting with the least such prime 53 that divide W(11) is (53, 79, 131). The sequence has 3 terms.

Gerard P. Michon, Factorization of Wendt's Determinant (table for n=1 to 114).
Eric Weisstein's World of Mathematics, Circulant matrix .
Wikipedia, Circulant matrix .

Cf. A035095, A048954.

w[n_] := Module[{x}, Resultant[x^n-1, (1+x)^n-1, x]]; k[n_, m_] := Module[{p=Prime@n, q=0, lst={}}, Do[q++; While[! PrimeQ[p*q+1], q++]; AppendTo[lst, q], {m}]; lst];
lst1 = {}; Do[lst=k[n, 50]*Prime[n]+1; m = 1; Do[If[IntegerQ[w[Prime[n]]/lst[[m]]]&&m<=Length@lst, m++, Break[]], {Length@lst}]; AppendTo[lst1, m-1], {n, 1, 75}]; lst1

A023503 Greatest prime divisor of prime(n) - 1.

Original entry on oeis.org

2, 2, 3, 5, 3, 2, 3, 11, 7, 5, 3, 5, 7, 23, 13, 29, 5, 11, 7, 3, 13, 41, 11, 3, 5, 17, 53, 3, 7, 7, 13, 17, 23, 37, 5, 13, 3, 83, 43, 89, 5, 19, 3, 7, 11, 7, 37, 113, 19, 29, 17, 5, 5, 2, 131, 67, 5, 23, 7, 47, 73, 17, 31, 13, 79, 11, 7, 173, 29, 11, 179, 61, 31, 7, 191

Offset: 2

Clark Kimberling

nonn

Baker & Harman (1998) show that there are infinitely many n such that a(n) > prime(n)^0.677. This improves on earlier work of Goldfeld, Hooley, Fouvry, Deshouillers, Iwaniec, Motohashi, et al.
Fouvry shows that a(n) > prime(n)^0.6683 for a positive proportion of members of this sequence. See Fouvry and also Baker & Harman (1996) which corrected an error in the former work.
The record values are the Sophie Germain primes A005384. - Daniel Suteu, May 09 2017
Conjecture: every prime is in the sequence. Cf. A035095 (see my comment). - Thomas Ordowski, Aug 06 2017
a(n) is 2 for n in A159611, and is at most 3 for n in A174099. Conjecture: liminf a(n) = 3. - Jeppe Stig Nielsen, Jul 04 2020

T. D. Noe, Table of n, a(n) for n = 2..10000
R. C. Baker and G. Harman, The Brun-Titchmarsh theorem on average, Analytic Number Theory (Proceedings in honor of Heini Halberstam), Birkhäuser, Boston, 1996, pp. 39-103.
R. Baker and G. Harman, Shifted primes without large prime factors, Acta Arithmetica 83 (1998), pp. 331-361.
Étienne Fouvry, Théorème de Brun-Titchmarsh; application au théorème de Fermat, Invent. Math 79 (1985), 383-407.
D. M. Goldfeld, On the number of primes p for which p + a has a large prime factor, Mathematika 16 (1969), pp. 23-27.
R. R. Hall, Some properties of the sequence {p-1}, Acta Arith. 28 (1975/76), 101-105.
G. Harman, On the greatest prime factor of p-1 with effective constants

Cf. A006093, A006530.

Cf. A005384, A035095, A159611, A174099.

A023503 := proc(n)
    A006530(ithprime(n)-1) ;
end proc:
seq( A023503(n),n=2..80) ; # R. J. Mathar, Sep 07 2016

Table[FactorInteger[Prime[n] - 1][[-1, 1]], {n, 2, 100}] (* T. D. Noe, Jun 08 2011 *)

a(n) = vecmax(factor(prime(n)-1)[,1]); \\ Michel Marcus, Aug 15 2015

a(n) = A006530(A006093(n)). - Michel Marcus, Aug 15 2015

Comments, references, and links from Charles R Greathouse IV, Mar 04 2011

A035096 a(n) is the smallest k such that prime(n)*k+1 is prime.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 6, 10, 2, 2, 10, 4, 2, 4, 6, 2, 12, 6, 4, 8, 4, 4, 2, 2, 4, 6, 6, 6, 10, 2, 4, 2, 6, 4, 8, 6, 10, 4, 14, 2, 2, 6, 2, 4, 18, 4, 10, 12, 24, 12, 2, 2, 6, 2, 6, 6, 8, 6, 4, 2, 6, 2, 4, 6, 6, 26, 6, 10, 6, 10, 14, 2, 6, 4, 12, 12, 24, 6, 8, 4, 2, 10, 2, 4, 10, 2, 8, 30

Offset: 1

Labos Elemer

nonn

These arithmetic progressions have prime differences. Note that both the terms of generated by this k values and the differences are primes as well.
This is one possible generalization of "the least prime problem in special arithmetic progressions" when n in the nk+1 form is replaced by n-th prime number.
Note that Dirichlet's theorem on primes in arithmetic progressions implies that a(n) always exists. - Max Alekseyev, Jul 11 2008
If a(n)=2, prime(n) is a Sophie Germain prime (A005384). Among the first 10^6 terms, the largest is a(330408) = 234. - Zak Seidov, Jan 28 2012

			a(15)=6 because the 15th prime is 47, and the smallest k such that 47k+1 is prime is k=6, for which 47k+1=283.

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Math, Dirichlet's theorem.
Index entries for sequences related to primes in arithmetic progressions.

Smallest k such that k*n+1 is prime is A034693.
Sophie Germain primes are in A005384.
Cf. A000040, A035095. - Zak Seidov, Dec 27 2013
Cf. A117673.

S:=[];
k:=1;
for n in [1..90] do
  while not IsPrime(k*NthPrime(n)+1) do
       k:=k+1;
  end while;
  Append(~S, k);
  k:=1;
end for;
S; // Bruno Berselli, Apr 18 2013

Reap[Sow[1]; Do[p = Prime[n]; k = 2; While[! PrimeQ[k*p + 1], k = k + 2]; Sow[k], {n, 2, 10^4}]][[2, 1]] (* Zak Seidov, Jan 28 2012 *)
f[n_] := Block[{p = Prime@ n}, q = 1 + 2p; While[ !PrimeQ@ q, q += 2p]; (q - 1)/p]; f[1] = 1; Array[f, 88] (* Robert G. Wilson v, Dec 27 2014 *)

a(n) = if(n == 1, 1, my(t = 2*prime(n), m = t + 1); while(!isprime(m), m += t); 2*(m - 1)/t); \\ Amiram Eldar, Mar 19 2025

a(n) = (A035095(n)-1)/A000040(n). - Zak Seidov, Dec 27 2013

A125878 Duplicate of A066674.

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

Artur Jasinski, Dec 13 2006

dead

Original name was: a(n) = the least number k such that cos(2pi/k) is an algebraic number of a prime(n)-smooth degree, but not prime(n-1)-smooth.
Comments from N. J. A. Sloane, Jan 07 2013: (Start)
This is a duplicate of A066674. This follows from the following argument. The degree of the minimal polynomial of cos(2*Pi/k) is phi(k)/2, where phi is Euler's totient function. Then a(n) is the least number k such that prime(n) is the largest prime dividing phi(k) and prime(n-1) does not divide phi(k)/2. For the rest of the proof see Bjorn Poonen's remarks in A066674.
It also seems likely that this is the same as A035095, but this is an open problem.
Conjecture: this sequence contains only primes (this would follow if this is indeed the same as A035095).
(End)

See A181877.

Cf. A066674, A035095, A125866-A125877.

Edited by Don Reble, Apr 24 2007

Minor edits by Ray Chandler, Oct 20 2011

A066674 Least number m such that phi(m) = A000010(m) is divisible by the n-th prime.

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, Dec 19 2001

nonn

All terms seem to be primes of the form a(n) = k*prime(n)+1 for some k.
Is this a duplicate of A035095? - R. J. Mathar, Dec 13 2008
For the first 5*10^6 terms, a(n) = A035095(n). - Donovan Johnson, Oct 21 2011
Comments on the relationship between A035095, A066674, A125878, added by N. J. A. Sloane, Jan 07 2013: (Start)
Let a(n) = A066674(n), b(n) = A035095(n), c(n) = A125878(n).
It is immediate from the definitions that a(n) <= b(n) and a(n) <= c(n).
Bjorn Poonen (Jan 06 2013) makes the following observations:
1) A prime p divides phi(m) if and only if p^2 | m or p | q-1 for some prime q | m. Thus the smallest m for p is either p^2 or the smallest prime q = 1 (mod p). In other words, a(n) = min(b(n),p(n)^2).
2) In particular, the m in the definition of a(n) is at most p(n)^2, so phi(m)/p(n) < p(n), so p(n) is the largest prime dividing phi(m), and phi(m)/(2 p(n)) < p(n)/2 < p(n-1), so p(n-1) does not divide phi(m)/2.
Thus c(n) = a(n).
Further comments from Eric Bach, Jan 07 2013: (Start)
As others have pointed out, the possible equivalence of a(n) and b(n) is basically the question of how quickly the least prime q == 1 mod p grows, as a function of p. In particular, if q < p^2, the two sequences are the same.
Here are some remarks connected with this.
1. There are probabilistic arguments suggesting that q = O(p (log p)^2). See Heath-Brown (1978), Wagstaff (1979), Bach and Huelsbergen (1993). Using the sieve of Eratosthenes, I found no exceptions to q < p^2 below p = 1254767. So it seems likely that a(n) and b(n) are the same.
2. If ERH holds, then q = O(p log p)^2, see Heath-Brown (1990), (1992). Explicitly, on the same hypothesis, q < 2(p log p)^2, see Bach and Sorenson (1996).
3. By Linnik's theorem, q = O(p^c) for some c > 0. This is unconditional, but the best known value of c, equal to 5.18 -- see Xylouris (2011) -- is nowhere near 2. Heath-Brown (1992) mentions the conjecture (generalized to Linnik's theorem) that q <= p^2. If true, a(n) and b(n) are identical, since p^2 cannot be 1 mod p. (End)
Don Reble (Jan 07 2013) observes that A074884 and A117673 are related to these questions.
Summary: A066674 and A125878 are the same, and A035095 is probably also the same, but this is an open question.
(End)

E. Bach and J. Shallit, Algorithmic Number Theory, Vol. 1: Efficient Algorithms, MIT Press, Cambridge, MA, 1996.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000
E. Bach and L. Huelsbergen, Statistical evidence for small generating sets, Math. Comp. 61 (1993) 69-82.
E. Bach and J. Sorenson, Explicit bounds for primes in residue classes, Math. Comp. 65, 1996, pp. 1717-1735.
D. R. Heath-Brown, Almost-primes in arithmetic progressions and short intervals, Math. Proc. Cambridge Phil. Soc., 83:357--375, 1978.
D. R. Heath-Brown, Siegel zeros and the least prime in an arithmetic progression, Quart. J. Math. Oxford (2) 41, 1990, 405-418.
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.
S. S. Wagstaff, Jr, Greatest of the Least Primes in Arithmetic Progressions Having a Given Modulus, Math. Comp., 33 (147) (1979) pp. 1073-1080.
T. Xylouris, On the least prime in an arithmetic progression and estimates for the zeros of Dirichlet L-functions, Acta Arith. 150 (2011), no. 1, 65-91.

Cf. A000010, A000040, A066675-A066678, A035095.

Cf. also A074884 and A117673.

f[n_] := Block[{m = p = Prime@ n}, While[ Mod[ EulerPhi@ m, p] != 0, m += 2]; m]; f[1] = 3; Array[f, 60] (* Robert G. Wilson v, Dec 27 2014 *)

a(n) = min{m : phi(m) = 0 mod prime(n) = 0}.

a(2) corrected by R. J. Mathar, Dec 13 2008

A068211 Largest prime factor of Euler totient function phi(n).

Original entry on oeis.org

2, 2, 2, 2, 3, 2, 3, 2, 5, 2, 3, 3, 2, 2, 2, 3, 3, 2, 3, 5, 11, 2, 5, 3, 3, 3, 7, 2, 5, 2, 5, 2, 3, 3, 3, 3, 3, 2, 5, 3, 7, 5, 3, 11, 23, 2, 7, 5, 2, 3, 13, 3, 5, 3, 3, 7, 29, 2, 5, 5, 3, 2, 3, 5, 11, 2, 11, 3, 7, 3, 3, 3, 5, 3, 5, 3, 13, 2, 3, 5, 41, 3, 2, 7, 7, 5, 11, 3, 3, 11, 5, 23, 3, 2, 3, 7, 5, 5

Offset: 3

Labos Elemer, Feb 21 2002

nonn

Smallest numbers m, such that largest prime factor of phi(m) is prime(n), a(n) is a prime number and identical to the n-th term of A035095: min{x: A068211(x) = prime(n)} = A035095(n). E.g., phi(A035095(7)) = phi(103) = 2*3*17 of which 17 = prime(7) is the largest prime factor.

			For n=46, phi(46) = 2*2*11, hence a(46) = 11.

T. D. Noe, Table of n, a(n) for n = 3..1000

Cf. A000010, A006530.

Cf. A035095, A035096.

[Maximum(PrimeDivisors(EulerPhi(n))): n in [3..90]]; // Vincenzo Librandi, Jan 04 2017

Table[FactorInteger[EulerPhi[n]][[-1, 1]], {n, 3, 100}] (* Vincenzo Librandi, Jan 04 2017 *)

a(n) = vecmax(factor(eulerphi(n))[,1]); \\ Michel Marcus, Jan 04 2017

a(n) = A006530(A000010(n)).

A129919 a(n) is the smallest prime of the form b*prime(n+1) + prime(n) with b > 0.

Original entry on oeis.org

5, 13, 19, 29, 37, 47, 131, 157, 139, 277, 179, 283, 127, 137, 577, 643, 181, 463, 919, 509, 389, 577, 439, 283, 1511, 307, 317, 761, 787, 367, 389, 953, 971, 1033, 3169, 1093, 809, 1499, 859, 3037, 541, 563, 577, 587, 1789, 2309, 1103, 677, 1601, 1627

Offset: 1

J. M. Bergot, Jun 05 2007

nonn

Dirichlet's theorem ensures that there always exists such a smallest prime because two primes are always coprime.

Corresponding values of b: 1,2,2,2,2,2,6,6,4, ... - Zak Seidov, Aug 29 2012

			a(2) is the smallest prime of the form 5*b+3 and b > 0. Hence a(2) = 13.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A035095.

a[n_] := Module[{k}, k = 1; While[Not[PrimeQ[k*Prime[n + 1] + Prime[n]]], k++ ]; k*Prime[n + 1] + Prime[n]]; Table[a[i], {i, 1, 50}]
sp[{a_,b_}]:=Module[{n=1},While[!PrimeQ[n*b+a],n++];n*b+a]; sp/@Partition[ Prime[Range[60]],2,1] (* Harvey P. Dale, Jan 02 2013 *)

Edited and extended by Stefan Steinerberger, Jul 16 2007

A032448 Smallest prime == -1 modulo prime(n).

Original entry on oeis.org

3, 2, 19, 13, 43, 103, 67, 37, 137, 173, 61, 73, 163, 257, 281, 211, 353, 487, 401, 283, 1021, 157, 331, 1423, 193, 1009, 617, 641, 653, 677, 761, 523, 547, 277, 1489, 1811, 313, 977, 1669, 691, 1789, 1447, 4201, 1543, 787, 397, 421, 1783, 907, 457

Offset: 1

Reinhard Zumkeller, Jun 25 2003

nonn

It appears that a(n) <= prime(n)^2-1, where prime(n) = A000040(n) is the n-th prime; see A035095 for a related conjecture. If correct, this implies A006530(a(n)+1)=prime(n), which in turn implies that there are no duplicated values in the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A035095, A088732, A006530, A000040.

a032448 n = head [q | q <- a000040_list, let p = a000040 n,
                      q `mod` p == p - 1]
-- Reinhard Zumkeller, Aug 08 2013

f[n_] := Block[{p = Prime@ n}, r = p - 1; While[ !PrimeQ@ r, r += p]; r]; Array[f, 50] (* Robert G. Wilson v, Jun 20 2014 *)

a(n) = {prn = prime(n); p = 2; while(p % prn != prn - 1, p = nextprime(p+1)); p;} \\ Michel Marcus, Aug 04 2013

Edited by Franklin T. Adams-Watters, Jun 21 2010

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.

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

A336280 Number of consecutive primes of the form k*prime(n) + 1, starting with the least such prime A035095(n), that divides the Wendt determinant A048954(prime(n)).

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A023503 Greatest prime divisor of prime(n) - 1.

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A035096 a(n) is the smallest k such that prime(n)*k+1 is prime.

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Magma

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A125878 Duplicate of A066674.

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A066674 Least number m such that phi(m) = A000010(m) is divisible by the n-th prime.

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Mathematica

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A068211 Largest prime factor of Euler totient function phi(n).

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Magma

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A129919 a(n) is the smallest prime of the form b*prime(n+1) + prime(n) with b > 0.

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