A005596 -id:A005596 - cp's OEIS frontend

A075432 Primes with no squarefree neighbors.

17, 19, 53, 89, 97, 127, 149, 151, 163, 197, 199, 233, 241, 251, 269, 271, 293, 307, 337, 349, 379, 449, 487, 491, 521, 523, 557, 577, 593, 631, 701, 727, 739, 751, 773, 809, 811, 881, 883, 919, 953, 991, 1013, 1049, 1051, 1061, 1063, 1097, 1151, 1171, 1249

Offset: 1

Reinhard Zumkeller, Sep 15 2002

nonn

Complement of A075430 in A000040.
From Ludovicus (luiroto(AT)yahoo.com), Dec 07 2009: (Start)
I propose a shorter name: non-Euclidean primes. That is justified by the Euclid's demonstration of the infinitude of primes. It appears that the proportion of non-Euclidean primes respect to primes tend to the limit 1-2A where A = 0.37395581... is Artin's constant. This table calculated by Jens K. Andersen corroborates it:
     10^5:        2421 /       9592 = 0.2523978315
     10^6:       19812 /      78498 = 0.2523885958
     10^7:      167489 /     664579 = 0.2520227091
     10^8:     1452678 /    5761455 = 0.2521373507
     10^9:    12817966 /   50847534 = 0.2520862860
     10^10:  114713084 /  455052511 = 0.2520875750
     10^11: 1038117249 / 4118054813 = 0.2520892256
It comes close to the expected 1-2A. (End)
This sequence is infinite by Dirichlet's theorem, since there are infinitely many primes == 17 or 19 (mod 36) and these have no squarefree neighbors. Ludovicus's conjecture about density is correct. Capsule proof: either p-1 or p+1 is divisible by 4, so it suffices to consider the other number (without loss of generality, p+1). For some fixed bound L, p is not divisible by any prime q < L (with finitely many exceptions) so there are q^2 - q possible residue classes for p. The primes in each are uniformly distributed so the probability that p+1 is divisible by q^2 is 1/(q^2 - q). The product of the complements goes to 2A as L increases without bound, and since 2A is an upper bound the limit is sandwiched between. - Charles R Greathouse IV, Aug 27 2014
Primes p such that both p-1 and p+1 are divisible by a square greater than 1. - N. J. A. Sloane, Jul 19 2024

			p = 17 is a term because 16 = 4^2 and 18=2*3^2 are divisible by squares > 1. - _N. J. A. Sloane_, Jul 19 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Pieter Moree, Artin's primitive root conjecture -a survey -, arXiv:math/0412262 [math.NT], 2004-2012.
Carlos Rivera, Conjecture 65. Non-Euclidean primes, The Prime Puzzles and Problems Connection.

Cf. A039787, A049097, A005117, A000040, A008966, A010051.

Intersection of A000040 and A281192.

a075432 n = a075432_list !! (n-1)
a075432_list = f [2, 4 ..] where
   f (u:vs@(v:ws)) | a008966 v == 1 = f ws
                   | a008966 u == 1 = f vs
                   | a010051' (u + 1) == 0 = f vs
                   | otherwise            = (u + 1) : f vs
-- Reinhard Zumkeller, May 04 2013

filter:= n -> isprime(n) and not numtheory:-issqrfree(n+1) and not numtheory:-issqrfree(n-1):
select(filter, [seq(2*i+1, i=1..1000)]); # Robert Israel, Aug 27 2014

lst={}; Do[p=Prime[n]; If[ !SquareFreeQ[Floor[p-1]] && !SquareFreeQ[Floor[p+1]], AppendTo[lst,p]], {n,6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 20 2008 *)
Select[Prime[Range[300]],!SquareFreeQ[#-1]&&!SquareFreeQ[#+1]&] (* Harvey P. Dale, Apr 24 2014 *)

is(n)=!issquarefree(if(n%4==1, n+1, n-1)) && isprime(n) \\ Charles R Greathouse IV, Aug 27 2014

a(n) ~ Cn log n, where C = 1/(1 - 2A) = 1/(1 - Product_{p>2 prime} (1 - 1/(p^2-p))), where A is the constant in A005596. - Charles R Greathouse IV, Aug 27 2014

More terms (that were already in the b-file) from Jeppe Stig Nielsen, Apr 23 2020

A088144 Sum of primitive roots of n-th prime.

Original entry on oeis.org

1, 2, 5, 8, 23, 26, 68, 57, 139, 174, 123, 222, 328, 257, 612, 636, 886, 488, 669, 1064, 876, 1105, 1744, 1780, 1552, 2020, 1853, 2890, 1962, 2712, 2413, 3536, 4384, 3335, 5364, 3322, 3768, 4564, 7683, 7266, 8235, 4344, 8021, 6176, 8274

Offset: 1

Ed Pegg Jr, Nov 03 2003

nonn

It is a result that goes back to Mirsky that the set of primes p for which p-1 is squarefree has density A, where A denotes the Artin constant (A = Product_{q prime} (1-1/(q*(q-1)))). Numerically A = 0.3739558136.. = A005596. More precisely, Sum_{p <= x} mu(p-1)^2 = Ax/log x + o(x/log x) as x tends to infinity. Conjecture: sum_{p <= x, mu(p-1)=1} 1 = (A/2)x/log x + o(x/log x) and sum_{p <= x, mu(p-1)=-1} 1 = (A/2)x/log x + o(x/log x). - Pieter Moree (moree(AT)mpim-bonn.mpg.de), Nov 03 2003
The number of the primitive roots is A008330(n). - R. K. Guy, Feb 25 2011
If prime(n)  == 1 (mod 4), then a(n) = prime(n)*A008330(n)/2. There are also primes of the form prime(n) == 3 (mod 4) where prime(n) | a(n), namely prime(n) = 19, 127, 151, 163, 199, 251,... The list of primes in both modulo-4 classes where prime(n)|a(n) is 5, 13, 17, 19, 29, 37, 41, 53, 61,... - R. K. Guy, Feb 25 2011
a(n) = A076410(n) at n = 1, 3, 7, 55,... (for p = 2, 5, 17, 257... and perhaps only for the Fermat primes). - R. K. Guy, Feb 25 2011

			For 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, the primitive roots are as follows: {{1}, {2}, {2, 3}, {3, 5}, {2, 6, 7, 8}, {2, 6, 7, 11}, {3, 5, 6, 7, 10, 11, 12, 14}, {2, 3, 10, 13, 14, 15}, {5, 7, 10, 11, 14, 15, 17, 19, 20, 21}, {2, 3, 8, 10, 11, 14, 15, 18, 19, 21, 26, 27}}

C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965; see p. 52.

T. D. Noe, Table of n, a(n) for n=1..1000
Leon Mirsky, The Number of Representations of an Integer as the Sum of a Prime and a k-Free Integer, Amer. Math. Monthly 56 (1949), 17-19.

Row sums of A060749, A254309.

Cf. A088145, A121380, A123475, A222009.

PrimitiveRootQ[ a_Integer, p_Integer ] := Block[ {fac, res}, fac = FactorInteger[ p - 1 ]; res = Table[ PowerMod[ a, (p - 1)/fac[ [ i, 1 ] ], p ], {i, Length[ fac ]} ]; ! MemberQ[ res, 1 ] ] PrimitiveRoots[ p_Integer ] := Select[ Range[ p - 1 ], PrimitiveRootQ[ #, p ] & ]
Total /@ Table[PrimitiveRootList[Prime[k]], {k, 1, 45}] (* Updated for Mathematica 13 by Harlan J. Brothers, Feb 27 2023 *)

a(n)=local(r, p, pr, j); p=prime(n); r=vector(eulerphi(p-1)); pr=znprimroot(p); for(i=1, p-1, if(gcd(i, p-1)==1, r[j++]=lift(pr^i))); vecsum(r) \\ after Franklin T. Adams-Watters's code in A060749, Michel Marcus, Mar 16 2015

A019335 Primes with primitive root 5.

Original entry on oeis.org

2, 3, 7, 17, 23, 37, 43, 47, 53, 73, 83, 97, 103, 107, 113, 137, 157, 167, 173, 193, 197, 223, 227, 233, 257, 263, 277, 283, 293, 307, 317, 347, 353, 373, 383, 397, 433, 443, 463, 467, 503, 523, 547, 557, 563, 577, 587, 593, 607, 613, 617, 647, 653, 673, 677, 683, 727

Offset: 1

David W. Wilson

nonn

To allow primes less than the specified primitive root m (here, 5) to be included, we use the essentially equivalent definition "Primes p such that the multiplicative order of m mod p is p-1". This comment applies to all of A019334-A019421. - N. J. A. Sloane, Dec 02 2019
Appears to be the numbers k such that the sequence 5^n mod k has period length k-1. All terms are congruent to 2 or 3 mod 5. - Gary Detlefs, May 21 2014
From Jianing Song, Apr 27 2019: (Start)
If we define
  Pi(N,b) = # {p prime, p <= N, p == b (mod 5)};
     Q(N) = # {p prime, p <= N, p in this sequence},
then by Artin's conjecture, Q(N) ~ (20/19)*C*N/log(N) ~ (40/19)*C*(Pi(N,2) + Pi(N,3)), where C = A005596 is Artin's constant.
Conjecture: if we further define
   Q(N,b) = # {p prime, p <= N, p == b (mod 5), p in this sequence},
then we have:
   Q(N,2) ~ (1/2)*Q(N) ~ (20/19)*C*Pi(N,2);
   Q(N,3) ~ (1/2)*Q(N) ~ (20/19)*C*Pi(N,3). (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Artin's constant
Wikipedia, Artin's conjecture on primitive roots
Index entries for primes by primitive root

Cf. A019334-A019421.

pr=5; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]

isok(p) = isprime(p) && (p != 5) && (znorder(Mod(5, p)) == p-1); \\ Michel Marcus, Apr 27 2019

A297408 Where the prime race among 10k+1, ..., 10k+9 changes leader.

Original entry on oeis.org

2, 13, 157, 193, 347, 383, 587, 673, 907, 1163, 1327, 1483, 1907, 1933, 2897, 4723, 5557, 5573, 6037, 6113, 6637, 6673, 7487, 8273, 8317, 8363, 8387, 8443, 8467, 8573, 8647, 8803, 8837, 8933, 9277, 9293, 10067, 10103, 11897, 11923, 12037, 12073, 12107, 12143

Offset: 1

Sean A. Irvine, Dec 29 2017

nonn

A007355 appears to be an erroneous version of this sequence.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993

Cf. A007352, A007350, A007353, A007354, A274121, A274122, A274123, A297406, A297407, A297408, A297410, A297411.

a297408(limit)={my(v=vector(10),vm=0,ivm=0,imv); forprime(p=2,limit,my(m=p%10);v[m]++;my(mv=vecmax(v,&imv));if(mv>vm,if(imv!=ivm,print1(p,", "); ivm=imv);vm=mv))};
a297408(12500) \\ Hugo Pfoertner, Jul 25 2021

from sympy import nextprime
from itertools import islice
def agen():
    c, p, leader = [0 for i in range(10)], 1, None
    while True:
        p = nextprime(p); last = p%10; c[last] += 1; m = max(c)
        if c.count(m) == 1 and c.index(m) == last and last != leader:
            yield p; leader = last
print(list(islice(agen(), 44))) # Michael S. Branicky, Dec 20 2022

A007353 Where the prime race among 5k+1, ..., 5k+4 changes leader.

Original entry on oeis.org

2, 83, 137, 293, 337, 443, 487, 523, 557, 743, 797, 1213, 1277, 1523, 1657, 1733, 1867, 1973, 2027, 2063, 2797, 2833, 2887, 4733, 5227, 5323, 5437, 5503, 5527, 5623, 5897, 5923, 6007, 6133, 6317, 6353, 6427, 6563, 6607, 6703, 7187, 7283, 7307, 7393, 7477, 8963

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

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..1000
Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993

Cf. A007352, A007350, A007353, A007354, A297406, A297407, A297408, A297410, A297411.

A007354 Where the prime race among 7k+1, ..., 7k+6 changes leader.

Original entry on oeis.org

2, 17, 131, 227, 733, 829, 929, 997, 1097, 1123, 1237, 1277, 1447, 1487, 1531, 1627, 1811, 1907, 1993, 2141, 2203, 2267, 2441, 2677, 2707, 3209, 3299, 3433, 3547, 3853, 4003, 4021, 4507, 4679, 4787, 4931, 5081, 5113, 7537, 7577, 7649, 7759, 7817, 8039, 8461, 8543

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

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..1000
Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993

Cf. A007352, A007350, A007353, A007354, A297406, A297407, A297408, A297410, A297411.

primerace(mod,limit)={my(v=vector(mod), vm=0, ivm=0, imv); forprime(p=2, limit, my(m=p%mod); v[m+1]++;  v[m+1]++; my(mv=vecmax(v, &imv)); if(mv>vm, if(imv!=ivm, print1(p, ", "); ivm=imv); vm=mv))};
primerace(7,8600) \\ Hugo Pfoertner, Jul 25 2021

A297406 Where the prime race among 8k+1, ..., 8k+7 changes leader.

Original entry on oeis.org

2, 11, 37, 83, 197, 227, 271, 293, 347, 373, 487, 547, 853, 907, 1069, 1447, 1733, 1831, 1949, 2027, 2053, 2131, 2237, 2251, 2309, 2719, 2749, 3019, 3061, 3083, 3733, 3779, 3877, 3931, 4919, 5179, 5303, 5347, 5407, 6661, 6911, 6949, 6967, 7459, 7789

Offset: 1

Sean A. Irvine, Dec 29 2017

Robert Israel, Table of n, a(n) for n = 1..10000
Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993

Cf. A007352, A007350, A007353, A007354, A297406, A297407, A297408, A297410, A297411.

X:= Vector(4): X[2]:= 1: m:= 3: p:= 3: count:= 1: R:= 2:
for i from 2 while count < 100 do
  p:= nextprime(p);
  j:= ((p mod 8)+1)/2;
  X[j]:= X[j]+1;
  mp:= max[index](X);
  if mp <> m and X[mp] > X[m] then
    m:= mp;
    R:= R,p;
    count:= count+1;
   fi
od:
R; # Robert Israel, Nov 21 2024

A297407 Where the prime race among 9k+1, ..., 9k+8 changes leader.

Original entry on oeis.org

2, 167, 191, 419, 461, 563, 587, 617, 677, 761, 857, 881, 929, 1427, 1451, 1607, 1667, 1777, 1823, 1867, 1913, 2351, 2399, 2459, 4127, 4583, 5039, 5087, 5171, 7283, 7349, 7517, 7547, 7643, 7691, 7901, 8681, 8837, 8933, 11243, 11903, 11927, 18329, 18371, 19913

Offset: 1

Sean A. Irvine, Dec 29 2017

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993

Cf. A007352, A007350, A007353, A007354, A297406, A297407, A297408, A297410, A297411.

V:= <(0$8)>:
lead:= 1:
Res:= NULL:
count:= 0:
p:= 1:
while count < 100 do
  p:= nextprime(p);
  t:= p mod 9;
  V[t]:= V[t]+1;
  if V[t] > V[lead] then
    lead:= t;
    Res:= Res, p;
    count:= count+1;
  fi
od:
Res; # Robert Israel, Jan 01 2018

A297410 Where the prime race among 11k+1, ..., 11k+10 changes leader.

Original entry on oeis.org

2, 73, 101, 149, 233, 359, 431, 509, 563, 1051, 1091, 1151, 1259, 1459, 1553, 1811, 2609, 2713, 2741, 4363, 4507, 4561, 4919, 5023, 5189, 6761, 7321, 7433, 7717, 7829, 8039, 8081, 8951, 9043, 9203, 9337, 9851, 9931, 10181, 10457, 11437, 11491, 13099, 19841

Offset: 1

Sean A. Irvine, Dec 29 2017

nonn

Pontus von Brömssen, Table of n, a(n) for n = 1..10000
Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993

Cf. A007352, A007350, A007353, A007354, A297406, A297407, A297408, A297410, A297411.

A297411 Where the prime race among 12k+1, ..., 12k+11 changes leader.

Original entry on oeis.org

2, 17, 79, 101, 163, 197, 211, 263, 281, 379, 401, 443, 461, 479, 631, 677, 739, 809, 907, 953, 1087, 1109, 1171, 1193, 1543, 1607, 1721, 1759, 2063, 2203, 2417, 2543, 2633, 2711, 2731, 2753, 3203, 3221, 3323, 3607, 3803, 3847, 3863, 3943, 4397

Offset: 1

Sean A. Irvine, Dec 29 2017

nonn

Pontus von Brömssen, Table of n, a(n) for n = 1..10000
Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993

Cf. A007352, A007350, A007353, A007354, A297406, A297407, A297408, A297410, A297411.

cp's OEIS Frontend

A075432 Primes with no squarefree neighbors.

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A088144 Sum of primitive roots of n-th prime.

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A019335 Primes with primitive root 5.

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A297408 Where the prime race among 10k+1, ..., 10k+9 changes leader.

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A007353 Where the prime race among 5k+1, ..., 5k+4 changes leader.

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A007354 Where the prime race among 7k+1, ..., 7k+6 changes leader.

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A297406 Where the prime race among 8k+1, ..., 8k+7 changes leader.

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A297407 Where the prime race among 9k+1, ..., 9k+8 changes leader.

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