A005596 -id:A005596 - cp's OEIS frontend

A307627 Primes p such that 2 is a primitive root modulo p while 8 is not.

13, 19, 37, 61, 67, 139, 163, 181, 211, 349, 373, 379, 421, 523, 541, 547, 613, 619, 661, 709, 757, 787, 829, 853, 859, 877, 883, 907, 1117, 1123, 1171, 1213, 1237, 1291, 1381, 1453, 1483, 1531, 1549, 1621, 1669, 1693, 1741, 1747, 1861, 1867, 1987, 2029, 2053

Offset: 1

Jianing Song, Apr 19 2019

nonn

Primes p such that 2 is a primitive root modulo p (i.e., p is in A001122) and that p == 1 (mod 3).

According to Artin's conjecture, the number of terms <= N is roughly ((2/5)*C)*PrimePi(N), where C is the Artin's constant = A005596, PrimePi = A000720. Compare: the number of terms of A001122 that are no greater than N is roughly C*PrimePi(N).

			For p = 67, the multiplicative order of 2 modulo 67 is 66, while 8^22 == 2^(3*22) == 1 (mod 67), so 67 is a term.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Artin's constant
Wikipedia, Artin's conjecture on primitive roots

Complement of A019338 with respect to A001122.

Cf. also A005596, A000720, A307628.

select(p -> isprime(p) and numtheory:-order(2,p) = p-1,
[seq(i,i=1..10000,6)]); # Robert Israel, Apr 23 2019

Select[Prime@ Range[5, 310], And[FreeQ[#, 8], ! FreeQ[#, 2]] &@ PrimitiveRootList@ # &] (* Michael De Vlieger, Apr 23 2019 *)

forprime(p=3, 2000, if(znorder(Mod(2, p))==(p-1) && p%3==1, print1(p, ", ")))

A307628 Primes p such that 2 is a primitive root modulo p while 32 is not.

Original entry on oeis.org

11, 61, 101, 131, 181, 211, 421, 461, 491, 541, 661, 701, 821, 941, 1061, 1091, 1171, 1291, 1301, 1381, 1451, 1531, 1571, 1621, 1741, 1861, 1901, 1931, 2131, 2141, 2221, 2371, 2531, 2621, 2741, 2851, 2861, 3011, 3371, 3461, 3491, 3571, 3581, 3691, 3701, 3851, 3931

Offset: 1

Jianing Song, Apr 19 2019

nonn

Primes p such that 2 is a primitive root modulo p (i.e., p is in A001122) and that p == 1 (mod 5).

By Artin's conjecture, the number of terms <= N is roughly ((4/19)*C)*PrimePi(N), where C is the Artin's constant = A005596, PrimePi = A000720. Compare: the number of terms of A001122 that are no greater than N is roughly C*PrimePi(N).

			For p = 61, the multiplicative order of 2 modulo 61 is 60, while 32^12 == 2^(5*12) == 1 (mod 61), so 61 is a term.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Artin's constant
Wikipedia, Artin's conjecture on primitive roots

Complement of A019358 with respect to A001122.

Cf. also A005596, A000720, A307627.

select(p -> isprime(p) and numtheory:-order(2,p) = p-1,
[seq(i,i=1..10000,10)]); # Robert Israel, Apr 23 2019

{11}~Join~Select[Prime@ Range[11, 550], And[FreeQ[#, 32], ! FreeQ[#, 2]] &@ PrimitiveRootList@ # &] (* Michael De Vlieger, Apr 23 2019 *)

forprime(p=3, 4000, if(znorder(Mod(2, p))==(p-1) && p%5==1, print1(p, ", ")))

A348971 a(n) = Product(p(p+1)^(e-1)) - Product((p-1)p^(e-1)), when n = Product(p^e), with p primes, and e their exponents.

Original entry on oeis.org

0, 1, 1, 4, 1, 4, 1, 14, 6, 6, 1, 14, 1, 8, 7, 46, 1, 18, 1, 22, 9, 12, 1, 46, 10, 14, 30, 30, 1, 22, 1, 146, 13, 18, 11, 60, 1, 20, 15, 74, 1, 30, 1, 46, 36, 24, 1, 146, 14, 40, 19, 54, 1, 78, 15, 102, 21, 30, 1, 74, 1, 32, 48, 454, 17, 46, 1, 70, 25, 46, 1, 192, 1, 38, 50, 78, 17, 54, 1, 238, 138, 42, 1, 102, 21

Offset: 1

Antti Karttunen, Nov 05 2021

Möbius transform of A348507.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A003959, A003968, A008683, A300251, A348507, A348970.

Cf. A005596, A104141.

f1[p_, e_] := p*(p + 1)^(e - 1); f2[p_, e_] := (p - 1)*p^(e - 1); a[1] = 0; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) - Times @@ f2 @@@ f; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)

A348971(n) = { my(f=factor(n),m1=1,m2=1,p); for(i=1, #f~, p = f[i, 1]; m1 *= p*(p+1)^(f[i, 2]-1); m2 *= (p-1)*p^(f[i, 2]-1)); (m1-m2); };

A348971(n) = { my(f=factor(n),p); for (i=1, #f~, p = f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f)-eulerphi(n); }

a(n) = A003968(n) - A000010(n).
a(n) = Sum_{d|n} A008683(n/d) * A348507(d).
Sum_{k=1..n} a(k) ~ c * n^2, where c = A104141 * (1/A005596 - 1) = 0.5088692487... . - Amiram Eldar, Oct 05 2023

A049232 Primes p such that p+2 is divisible by a square.

Original entry on oeis.org

2, 7, 23, 43, 47, 61, 73, 79, 97, 151, 167, 173, 223, 241, 277, 313, 331, 349, 359, 367, 373, 421, 439, 457, 523, 547, 601, 619, 673, 691, 709, 727, 733, 773, 823, 839, 853, 907, 929, 997, 1033, 1051, 1069, 1087, 1123, 1181, 1213, 1223, 1231, 1249, 1303

Offset: 1

Labos Elemer

nonn

This sequence is infinite and its relative density in the sequence of the primes is equal to 1 - 2 * Product_{p prime} (1-1/(p*(p-1))) = 1 - 2 * A005596 = 0.252088... - Amiram Eldar, Feb 27 2021

			47 is a term since 47+2 = 49 = 7^2 is a square.
523 is a term since 523+2 = 525 = 5^2*21 is divisible by a square.

A091880 gives prime indices.

Cf. A005596, A013929, A049229, A049233.

Select[Prime[Range[100]], ! SquareFreeQ[ # + 2] &] (* Zak Seidov, Oct 28 2008 *)

powerfreep3(n,p,k) = { c=0; pc=0; forprime(x=2,n, pc++; if(ispowerfree(x+k,p)==0, c++; print1(x","); ) ); print(); print(c","pc","c/pc+.0) }
ispowerfree(m,p1) = { flag=1; y=component(factor(m),2); for(i=1,length(y), if(y[i] >= p1,flag=0;break); ); return(flag) }

Primes p such that mu(p+2) = 0.

Corrected by Cino Hilliard and Ray Chandler, Dec 08 2003

Edited by N. J. A. Sloane, Oct 27 2008 at the suggestion of R. J. Mathar

A105880 Primes for which -8 is a primitive root.

Original entry on oeis.org

5, 23, 29, 47, 53, 71, 101, 149, 167, 173, 191, 197, 239, 263, 269, 293, 311, 317, 359, 383, 389, 461, 479, 503, 509, 557, 599, 647, 653, 677, 701, 719, 743, 773, 797, 821, 839, 863, 887, 941, 983, 1031, 1061, 1109, 1151, 1223, 1229, 1277, 1301, 1319, 1367, 1373, 1439

Offset: 1

N. J. A. Sloane, Apr 24 2005

nonn

From Jianing Song, May 12 2024: (Start)
Members of A105874 that are not congruent to 1 mod 3. Terms are congruent to 5 or 23 modulo 24.
According to Artin's conjecture, the number of terms <= N is roughly ((3/5)*C)*PrimePi(N), where C is the Artin's constant = A005596, PrimePi = A000720. Compare: the number of terms of A001122 that are no greater than N is roughly C*PrimePi(N). (End)

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

Cf. A105874, A005596, A000720.

pr=-8; Select[Prime[Range[400]], MultiplicativeOrder[pr, # ] == #-1 &] (* N. J. A. Sloane, Jun 01 2010 *)
a[p_, q_]:= Sum[2 Cos[2^n Pi/((2 q+1)(2 p+1))],{n,1,2 q p}]
2 Select[Range[800], Rationalize[N[a[#, 9], 20]] == 1 &] + 1
(* Gerry Martens, Apr 28 2015 *)

is(n)=isprime(n) && n>3 && znorder(Mod(-8,n))==n-1 \\ Charles R Greathouse IV, May 21 2015

Let a(p,q)=sum(n=1,2*p*q,2*cos(2^n*Pi/((2*q+1)*(2*p+1)))). Then 2*p+1 is a prime of this sequence when a(p,9)==1. - Gerry Martens , May 21 2015

A192862 Flat primes: odd primes p such that p+1 is a squarefree number times a power of two.

Original entry on oeis.org

3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 41, 43, 47, 59, 61, 67, 73, 79, 83, 101, 103, 109, 113, 127, 131, 137, 139, 151, 157, 163, 167, 173, 181, 191, 193, 211, 223, 227, 229, 239, 257, 263, 271, 277, 281, 283, 307, 311, 313, 317, 331, 347, 353, 367, 373, 379

Offset: 1

Charles R Greathouse IV, Jul 11 2011

nonn

Broughan & Qizhi show that this sequence has relative density 2A in the primes, where A = A005596 is Artin's constant. Consequently, there exists a flat number between x and (1+e)x for every e > 0 and large enough x.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Kevin Broughan and Zhou Qizhi, Flat primes and thin primes, Bulletin of the Australian Mathematical Society 82:2 (2010), pp. 282-292.
Qizhi Zhou, Multiply perfect numbers of low abundancy, PhD thesis (2010)

Subsequence of A192861.

Cf. A192863, A192864, A005596.

is(n)=issquarefree((n+1)>>valuation(n+1,2)) && isprime(n) && n>2 \\ Charles R Greathouse IV, Mar 26 2014

a(n) ~ k * n * log(n) with k = 1/(2A) = 1.3370563...

A221981 Primes q = 4*p+1, where p == 2 (mod 5) is also prime.

Original entry on oeis.org

29, 149, 269, 389, 509, 1109, 1229, 1949, 2309, 2909, 3989, 4349, 5189, 5309, 6269, 6389, 7109, 7949, 8069, 9749, 10589, 10709, 11069, 11549, 12149, 12269, 13229, 13829, 14549, 15629, 16229, 17189, 17789, 18269, 19949, 20789, 22109, 22229, 24029, 24989, 25349, 25469, 25589, 26189, 26309, 28109, 28229, 28949, 29669, 30029, 30869, 31469, 32069, 33149, 34589, 34949, 36269, 36629, 36749, 37589

Offset: 1

Jonathan Sondow, Feb 02 2013

nonn

Moree (2012) says that Chebyshev observed that if q = 4p + 1 is prime, with prime p == 2 (mod 5), then 10 is a primitive root modulo q.
If the sequence is infinite, then Artin's conjecture ("every nonsquare integer n != -1 is a primitive root of infinitely many primes q") is true for n = 10.
The corresponding primes p are A221982.
The sequence is infinite under Dickson's conjecture, thus Dickson's conjecture implies Artin's conjecture for n = 10. - Charles R Greathouse IV, Apr 18 2013
Two conjectures: (a) These primes have primitive root 40; (b) if a(n)*8 + 1 is prime then it has primitive root 10. - Davide Rotondo, Dec 31 2024

			29 is a member because 29 = 4*7 + 1 and 7 == 2 (mod 5) are prime.

P. L. Chebyshev, Theory of congruences, Elements of number theory, Chelsea, 1972, p. 306.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section F9, pp. 377-380.

Amiram Eldar, Table of n, a(n) for n = 1..10000
P. L. Chebyshev, Theorie der Congruenzen, Mayer & Mueller, 1889, pp. 306-313.
Pieter Moree, Artin's primitive root conjecture - a survey, arXiv:math/0412262 [math.NT], 2004, revised 2012, p. 1.
Index entries for sequences related to Artin's conjecture
Index entries for primes by primitive root

Cf. A001913, A005596, A006883, A045380, A106849, A221982, A222008.

A221981:=n->`if`(isprime(4*n+1) and isprime(n) and n mod 5 = 2, 4*n+1, NULL): seq(A221981(n), n=1..10^4); # Wesley Ivan Hurt, Dec 11 2015

Select[ Prime[ Range[4000]], Mod[(# - 1)/4, 5] == 2 && PrimeQ[(# - 1)/4] &]

is(n)=n%20==9 && isprime(n) && isprime(n\4) \\ Charles R Greathouse IV, Apr 18 2013

a(n) = 4*A221982(n) + 1.

a(n) >> n log^2 n. - Charles R Greathouse IV, Dec 30 2024

A222008 Primes of the form 4^k + 1 for some k > 0.

Original entry on oeis.org

5, 17, 257, 65537

Offset: 1

Jonathan Sondow, Feb 04 2013

nonn

Same as Fermat primes 2^(2^m) + 1 for m >= 1. See A019434 for comments, etc.
Chebyshev showed that 3 is a primitive root of all primes of the form 2^(2*k) + 1 with k > 0. If the sequence is infinite, then Artin's conjecture ("every nonsquare integer n != -1 is a primitive root of infinitely many primes q") is true for n = 3.
As a(n) is a Fermat prime > 3, by Pépin's test a(n) has primitive root 3.
Conjecture: let p a prime number, a(n) is not congruent to p mod (p^2-3)/2. - Vincenzo Librandi, Jun 15 2014
This conjecture is false when p = a(n), but may be true for primes p != a(n). - Jonathan Sondow, Jun 15 2014
Primes p with the property that k-th smallest divisor of its squares p^2, for all 1 <= k <= tau(p^2), contains exactly k "1" digits in its binary representation. Corresponding values of squares p^2: 25, 289, 66049, 4295098369. Example: p = 257, set of divisors of p^2 = 66049 in binary representation: {1, 100000001, 10000001000000001}. Subsequence of A255401. - Jaroslav Krizek, Dec 21 2016

			4^1 + 1 = 5 is prime, so a(1) = 5. Also, 3^k == 3, 4, 2, 1 (mod 5) for k = 1, 2, 3, 4, resp., so 3 is a primitive root for a(1).

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966, p. 102, nr. 3.
P. L. Chebyshev, Theory of congruences. Elements of number theory, Chelsea, 1972, p. 306.

P. L. Chebyshev, Theorie der Congruenzen, Mayer & Mueller, 1889, p. 306-313.
R. Fueter, Über primitive Wurzeln von Primzahlen, Comment. Math. Helv., 18 (1946), 217-223, p. 217.
Wikipedia, Pépin's test
Index entries for sequences related to Artin's conjecture
Index entries for primes by primitive root

Cf. A005596, A019334, A019434, A090866, A221981.

Select[Table[4^k + 1, {k, 10^3}], PrimeQ] (* Michael De Vlieger, Dec 22 2016 *)

a(n) = A019434(n+1) for n > 0.

A307410 Numerators of the product in the singular series.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 5, 1, 1, 3, 9, 1, 11, 5, 3, 1, 15, 1, 17, 3, 5, 9, 21, 1, 3, 11, 1, 5, 27, 3, 29, 1, 9, 15, 5, 1, 35, 17, 11, 3, 39, 5, 41, 9, 3, 21, 45, 1, 5, 3, 15, 11, 51, 1, 27, 5, 17, 27, 57, 3, 59, 29, 5, 1, 11, 9, 65, 15, 21, 5, 69, 1, 71, 35, 3, 17, 3, 11, 77, 3, 1, 39, 81, 5, 45

Offset: 1

Mats Granvik, Apr 07 2019

Differs from A305444 at n = 35, 65, 70, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
John Omielan, How do you compute the singular series?, Mathematics Stack Exchange.
Terence Tao, Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges. See the next formula after equation 2.

Cf. A005596, A005597, A305444, A380839 (denominators).

f:= proc(n) numer(mul((p-2)/(p-1),p=select(type,numtheory:-factorset(n),odd))) end proc:
map(f, [$1..100]); # Robert Israel, Apr 07 2019

Table[Times@@(DeleteDuplicates[DeleteCases[DeleteCases[Exp[MangoldtLambda[Divisors[h]]], 1],2]] - 2)/Times@@(DeleteDuplicates[DeleteCases[DeleteCases[Exp[MangoldtLambda[Divisors[h]]], 1], 2]] - 1), {h, 1, 85}]
Numerator[%]
f[p_, e_] := If[p == 2, 1, (p-2)/(p-1)]; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Mar 03 2025 *)

a(n) = my(f=factor(n)[,1]~); numerator(prod(k=1, #f, if (f[k]>2, (f[k]-2)/(f[k]-1), 1))); \\ Michel Marcus, Apr 07 2019

a(n) = numerator of Product_{p|n;p>2}(p-2)/(p-1) where p is a prime number.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A380839(k) = 2 * Product_{p prime} (1-1/(p^2-p)) = 2 * A005596 = 0.7479116272384045761094... . - Amiram Eldar, Mar 03 2025

A323594 Primes p such that 3 is a primitive root modulo p while 27 is not.

Original entry on oeis.org

7, 19, 31, 43, 79, 127, 139, 163, 199, 211, 223, 283, 331, 379, 463, 487, 571, 607, 631, 691, 739, 751, 811, 823, 859, 907, 1039, 1063, 1087, 1123, 1231, 1279, 1291, 1327, 1423, 1447, 1459, 1483, 1567, 1579, 1627, 1663, 1699, 1723, 1747, 1831, 1951, 1987, 1999

Offset: 1

Jianing Song, Aug 30 2019

nonn

Primes p such that 3 is a primitive root modulo p (i.e., p is in A019334) and that p == 1 (mod 3).

According to Artin's conjecture, the number of terms <= N is roughly ((2/5)*C)*PrimePi(N), where C is the Artin's constant = A005596, PrimePi = A000720. Compare: the number of terms of A001122 that are no greater than N is roughly C*PrimePi(N).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Artin's constant.
Wikipedia, Artin's conjecture on primitive roots.

Complement of A019353 with respect to A019334.
Cf. also A005596, A000720.
Primes p such that 3 is a primitive root modulo p and that p == 1 (mod q): this sequence (q=3), A323617 (q=5), A323628 (q=7).

forprime(p=5, 2000, if(znorder(Mod(3, p))==(p-1) && p%3==1, print1(p, ", ")))

cp's OEIS Frontend

A307627 Primes p such that 2 is a primitive root modulo p while 8 is not.

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A307628 Primes p such that 2 is a primitive root modulo p while 32 is not.

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A348971 a(n) = Product(p*(p+1)^(e-1)) - Product((p-1)*p^(e-1)), when n = Product(p^e), with p primes, and e their exponents.

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A049232 Primes p such that p+2 is divisible by a square.

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A105880 Primes for which -8 is a primitive root.

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A192862 Flat primes: odd primes p such that p+1 is a squarefree number times a power of two.

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A221981 Primes q = 4*p+1, where p == 2 (mod 5) is also prime.

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A348971 a(n) = Product(p(p+1)^(e-1)) - Product((p-1)p^(e-1)), when n = Product(p^e), with p primes, and e their exponents.