A005596 -id:A005596 - cp's OEIS frontend

A120629 Numbers k with property that -k is not a perfect power and the squarefree part of -k is not congruent to 1 modulo 4.

2, 4, 5, 6, 9, 10, 13, 14, 16, 17, 18, 20, 21, 22, 24, 25, 26, 29, 30, 33, 34, 36, 37, 38, 40, 41, 42, 45, 46, 49, 50, 52, 53, 54, 56, 57, 58, 61, 62, 65, 66, 68, 69, 70, 72, 73, 74, 77, 78, 80, 81, 82, 84, 85, 86, 88, 89, 90, 93, 94, 96, 97, 98, 100, 101, 102, 104, 105, 106

Offset: 1

Gerard P. Michon, Jun 20 2006

According to a famous 1927 conjecture of Emil Artin, modified by Dick Lehmer, these negative numbers are primitive roots modulo each prime of a set whose density among primes equals Artin's constant (see A005596). The positive numbers with the same property are given by A085397.

			-3 and -12 are not in the set because their squarefree parts are equal to -3, which is congruent to 1 modulo 4. -32 is not in the set because it is the fifth power of -2. -1 is excluded because it is an odd power of -1.

T. D. Noe, Table of n, a(n) for n = 1..1000
G. P. Michon, Artin's Constant.

Cf. A085397, A005596.

SquareFreePart[n_] := Times @@ Apply[ Power, ({#[[1]], Mod[#[[2]], 2]} & ) /@ FactorInteger[n], {1}]; perfectPowerQ[n_] := (r = False; For[k = 2, k <= Abs[n] + 2, k++, If[Reduce[n == x^k, {x}, Integers] =!= False, r = True; Break[]]]; r); ok[n_] := ! perfectPowerQ[-n] && Mod[SquareFreePart[-n], 4] != 1; Select[Range[106], ok](* Jean-François Alcover, Feb 14 2012 *)

A146481 Decimal expansion of Product_{n>=2} (1 - 1/(n*(n-1))).

Original entry on oeis.org

2, 9, 6, 6, 7, 5, 1, 3, 4, 7, 4, 3, 5, 9, 1, 0, 3, 4, 5, 7, 0, 1, 5, 5, 0, 2, 0, 2, 1, 9, 1, 4, 2, 8, 6, 4, 8, 6, 4, 8, 3, 1, 5, 1, 9, 1, 7, 8, 9, 4, 7, 8, 9, 0, 8, 1, 6, 7, 3, 5, 7, 3, 3, 1, 6, 5, 9, 0, 6, 1, 6, 2, 9, 1, 5, 1, 9, 6, 0, 8, 8, 8, 3, 6, 6, 7, 4, 8, 1, 6, 4, 0, 2, 1, 2, 6, 2, 2, 1, 4, 5, 4, 1, 7, 7

Offset: 0

R. J. Mathar, Feb 13 2009

Product of Artin's constant A005596 and the equivalent almost-prime products.

			0.2966751347435910345... = (1 - 1/2)*(1 - 1/6)*(1 - 1/12)*(1 - 1/20)*...

M. Chamberland, A. Straub, On gamma constants and infinite products, arXiv:1309.3455
R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], first line Table 3.

Cf. A005596.

phi := (1+sqrt(5))/2; evalf(-sin(Pi*phi)/Pi) ; # R. J. Mathar, Feb 20 2009

RealDigits[-Cos[Pi*Sqrt[5]/2]/Pi, 10, 105] // First (* Jean-François Alcover, Feb 11 2013 *)

The logarithm is -Sum_{s>=2} Sum_{j=1..floor(s/(1+r))} binomial(s-r*j-1, j-1)*(1-Zeta(s))/j at r=1.
s*Sum_{j=1..floor(s/2)} binomial(s-j-1, j-1)/j = A001610(s-1).
Equals 1/Product_{k=1..2} Gamma(1-x_k) = -sin(A094886)/A000796, where x_k are the 2 roots of the polynomial x*(x+1)-1. [R. J. Mathar, Feb 20 2009]

More terms from Jean-François Alcover, Feb 11 2013

A146482 Decimal expansion of Product_{q in A001358} (1-1/(q*(q-1))).

Original entry on oeis.org

8, 3, 9, 0, 4, 2, 1, 5, 4, 2, 7, 4, 4, 6, 8, 6, 0, 0, 7, 6, 8, 4, 6, 2, 1, 1, 1, 1, 9, 4, 5, 4, 1, 2, 5, 4, 9, 2, 8, 3, 0, 7, 1, 6, 6, 7, 6, 0, 8, 8, 2, 7, 3, 3, 0, 0, 0

Offset: 0

R. J. Mathar, Feb 13 2009

Semiprime analog of A005596.

			0.839042154274468600768... = (1-1/12)*(1-1/30)*(1-1/72)*(1-1/90)*(1-1/182)*..

R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], Table 3 third line. [From _R. J. Mathar_, Mar 28 2009]

The logarithm is -sum_{s>=2} sum_{j=1..floor[s/(1+r)]} binomial(s-r*j-1,j-1)*P_2(s)/j at r=1, where P_k(s) are the k-almost prime zeta functions of arXiv:0803.0900.

A146483 Decimal expansion of Product_{q in A014612} (1-1/(q*(q-1))).

Original entry on oeis.org

9, 5, 8, 7, 5, 2, 1, 1, 6, 4, 3, 5, 7, 3, 0, 9, 2, 7, 7, 1, 4, 7, 4, 0, 2, 5, 6, 5, 7, 8, 9, 2, 8, 6, 1, 2, 6, 5, 9, 4, 9, 0, 4, 4, 8, 5, 0, 2, 3, 5, 9, 9, 0, 1, 5, 9, 2

Offset: 0

R. J. Mathar, Feb 13 2009

3-almost prime analog of A005596.

			0.9587521164357309277147402... = (1-1/56)*(1-1/132)*(1-1/306)*(1-1/380)*..

R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], table 3 fourth line. [From _R. J. Mathar_, Mar 28 2009]

The logarithm is -sum_{s>=2} sum_{j=1..floor[s/(1+r)]} binomial(s-r*j-1,j-1)*P_3(s)/j at r=1, where P_k(s) are the k-almost prime zeta functions of arXiv:0803.0900.

A146484 Decimal expansion of Product_{q in A014613} (1-1/(q*(q-1))).

Original entry on oeis.org

9, 8, 9, 6, 2, 8, 8, 6, 7, 1, 6, 6, 4, 2, 7, 6, 6, 5, 5, 0, 4, 3, 2, 2, 8, 3, 7, 4, 5, 7, 9, 2, 4, 3, 0, 8, 0, 5, 7, 5, 5, 7, 5, 8, 9, 3, 5, 0, 2, 9, 6, 5, 3, 4, 8, 4, 4

Offset: 0

R. J. Mathar, Feb 13 2009

4-almost prime analog of A005596.

			0.989628867166427665504.. = (1-1/240)*(1-1/552)*...

The logarithm is -sum_{s>=2} sum_{j=1..floor[s/(1+r)]} binomial(s-r*j-1,j-1)*P_4(s)/j at r=1, where P_k(s) are the k-almost prime zeta functions of arXiv:0803.0900.

A271869 Decimal expansion of Matthews' constant C_3, an analog of Artin's constant for primitive roots.

Original entry on oeis.org

0, 6, 0, 8, 2, 1, 6, 5, 5, 1, 2, 0, 3, 0, 5, 0, 8, 6, 0, 0, 5, 6, 3, 2, 2, 7, 5, 4, 6, 1, 9, 2, 0, 8, 5, 5, 4, 3, 1, 3, 3, 7, 3, 7, 3, 4, 7, 5, 7, 6, 7, 9, 4, 1, 9, 8, 2, 6, 4, 3, 4, 0, 3, 1, 5, 0, 4, 0, 8, 0, 4, 3, 5, 0, 7, 2, 1, 2, 5, 6, 1, 6, 9, 5, 8, 6, 1, 8, 8, 8, 7, 3, 4, 8, 5, 8, 6, 6, 2, 4, 6, 8, 7, 3, 4, 0

Offset: 0

Jean-François Alcover, Apr 16 2016

			0.0608216551203050860056322754619208554313373734757679419826434...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.4 Artin's constant, p. 105.

K. R. Matthews, A generalisation of Artin's conjecture for primitive roots, Acta arithmetica, Vol. 29, No. 2 (1976), pp. 113-146.

Cf. A005596, A065414, A065415, A065416, A271780, A271798.

digits = 70; $MaxExtraPrecision = 1000; m0 = 2000; dm = 200; Clear[s]; LR =
LinearRecurrence[{2, 2, -6, 4, -1}, {0, 6, 0, 22, 5}, 2 m0]; r[n_Integer] := LR[[n]]; s[m_] := s[m] = NSum[-r[n] PrimeZetaP[n]/n, {n, 2, m}, NSumTerms -> 2 m0, WorkingPrecision -> digits+10] // Exp; s[m0]; s[m = m0+dm]; While[RealDigits[s[m], 10, digits][[1]] != RealDigits[ s[m-dm], 10, digits][[1]], Print[m]; m = m + dm]; Join[{0}, RealDigits[ s[m], 10, digits][[1]]]

prodeulerrat(1 - (p^3 - (p - 1)^3)/(p^3*(p - 1))) \\ Amiram Eldar, Mar 16 2021

C_3 = Product_{p prime} 1 - (p^3 - (p - 1)^3)/(p^3*(p - 1)).

More digits from Vaclav Kotesovec, Jun 19 2020

A306198 Multiplicative with a(p^e) = p^(e-1)*(p^2 - p - 1).

Original entry on oeis.org

1, 1, 5, 2, 19, 5, 41, 4, 15, 19, 109, 10, 155, 41, 95, 8, 271, 15, 341, 38, 205, 109, 505, 20, 95, 155, 45, 82, 811, 95, 929, 16, 545, 271, 779, 30, 1331, 341, 775, 76, 1639, 205, 1805, 218, 285, 505, 2161, 40, 287, 95, 1355, 310, 2755, 45, 2071, 164, 1705, 811

Offset: 1

Jianing Song, Jan 28 2019

For any positive integer n and any m coprime to n, define R(n,m) = Product_{primes p divides n} (p - [m == 1 (mod p)]), where [] is an Iverson branket. Then we have the following conjecture: (Start)
Let k == 2, 3 (mod 4) be a squarefree number, b be any positive integer such that k*b^2 is not a perfect power and not equal to -1, n be either coprime to or divisible by 4*k. Define Q(N,k*b^2,n,m) = # {primes p <= N : p == m (mod n), k*b^2 is a primitive modulo p}, then:
(a) if gcd(n, 4*k) = 1, then Q(N,k*b^2,n,m)/(C*PrimePi(N)) ~ R(n,m)/a(n);
(b) if 4*k divides n, then Q(N,k*b^2,n,m)/(C*PrimePi(N)) ~ 2*R(n,m)/a(n) if Jacobi(k/m) = -1 and 0 if Jacobi(k/m) = +1,
Where C is the Artin's constant = A005596, PrimePi = A000720. (End)
(Note that Sum_{m=1..n, gcd(m,n)=1} R(n,m) = a(n).)
For example, let N = 10^6:
k*b^2 |  n |  m | Q(N,k*b^2,n,m) | Q(N,k*b^2,n,m)/(C*PrimePi(N))
   2  |  8 |  3 |      14642     |   0.498794... approx = 2/4
   3  |  5 |  1 |       6192     |   0.210936... approx = 4/19
  -2  | 48 | 13 |       2933     |   0.099915... approx = 4/40
  -5  |  9 |  5 |       5933     |   0.202113... approx = 3/15

Jianing Song, 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.

Cf. A000720 (PrimePi), A005596 (Artin's constant), A086463.

P := (p, e) -> p^(e-1)*(p^2 - p - 1):
a := n -> mul(P(f[1], f[2]), f in ifactors(n)[2]):
seq(a(n), n=1..58); # Peter Luschny, Feb 13 2019

a[n_] := Product[{p, e} = pe; p^(e-1) (p^2-p-1), {pe, FactorInteger[n]}]; a[1] = 1; Array[a, 58] (* Jean-François Alcover, Jul 22 2019 *)

a(n) = my(f=factor(n)); prod(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]); (p^2 - p - 1)*p^(e-1))

Sum_{k=1..n} a(k) ~ c * n^3, where c = (Pi^2/18) * Product_{p prime} (1 - 3/p^2 + 1/p^3 + 1/p^4) = 0.1314639252... . - Amiram Eldar, Dec 01 2022

A333315 a(n) = Sum_{k=1..n} phi(prime(k)-1), where phi is the Euler totient function (A000005).

Original entry on oeis.org

1, 2, 4, 6, 10, 14, 22, 28, 38, 50, 58, 70, 86, 98, 120, 144, 172, 188, 208, 232, 256, 280, 320, 360, 392, 432, 464, 516, 552, 600, 636, 684, 748, 792, 864, 904, 952, 1006, 1088, 1172, 1260, 1308, 1380, 1444, 1528, 1588, 1636, 1708, 1820, 1892, 2004, 2100, 2164

Offset: 1

Amiram Eldar, Mar 14 2020

nonn

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 30.

Amiram Eldar, Table of n, a(n) for n = 1..10000
S. S. Pillai, On the sum function connected with primitive roots, Proceedings of the Indian Academy of Sciences - Section A, Vol. 13 (1941), pp. 526-529, alternative link.
Eric Weisstein's World of Mathematics, Logarithmic Integral.
Wikipedia, Logarithmic integral function.

Partial sums of A008330.

Cf. A000005, A005596.

Accumulate @ EulerPhi[Select[Range[300], PrimeQ] - 1]

a(n) = sum(k=1, n, eulerphi(prime(k)-1)); \\ Michel Marcus, Mar 15 2020

a(n) = Sum_{k=1..n} A008330(k).

a(n) ~ A * Li(n^2), where A is Artin's constant (A005596), and Li(x) is the logarithmic integral function.

A340154 Primes p such that p == 5 (mod 6) and p+1 is squarefree.

Original entry on oeis.org

5, 29, 41, 101, 113, 137, 173, 257, 281, 317, 353, 389, 401, 461, 509, 569, 617, 641, 653, 677, 761, 797, 821, 857, 929, 941, 977, 1109, 1181, 1193, 1217, 1229, 1289, 1301, 1361, 1373, 1409, 1433, 1481, 1553, 1613, 1697, 1721, 1877, 1901, 1913, 1973, 2081, 2129

Offset: 1

Amiram Eldar, Dec 29 2020

Clary and Fabrykowski (2004) proved that this sequence is infinite, and that its relative density in the sequence of primes of the form 6*k+5 (A007528) is 4*A/5 = 0.29916465..., where A is Artin's constant (A005596).

			5 is a term since it is prime, 5 == 5 (mod 6), and 5+1 = 6 = 2*3 is squarefree.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Stuart Clary and Jacek Fabrykowski, Arithmetic progressions, prime numbers, and squarefree integers, Czechoslovak Mathematical Journal, Vol. 54, No. 4 (2004), pp. 915-927.

Intersection of A007528 and A049097.

Cf. A005117, A005596.

Select[Range[2000], Mod[#, 6] == 5 && PrimeQ[#] && SquareFreeQ[# + 1] &]

A351682 Prime numbers p such that the (p-1)-st Bell number B(p-1) is a primitive root modulo p.

Original entry on oeis.org

2, 3, 11, 13, 17, 19, 29, 31, 47, 53, 71, 103, 113, 127, 131, 137, 139, 149, 173, 179, 181, 191, 211, 233, 239, 241, 251, 257, 263, 269, 293, 317, 347, 367, 379, 401, 431, 439, 449, 461, 503, 509, 523, 541, 557, 587, 607, 617, 619, 647, 653, 683, 691, 733, 743, 761, 773, 797, 821, 823, 827, 853, 859, 881, 919, 929

Offset: 1

Luis H. Gallardo, May 04 2022

nonn

Heuristically, the density of the sequence in the primes should approach Artin's constant: 0.3739558136...

			For n = 2 one has a(2) = 3 since B(2) = 2 is a primitive root modulo 3.

Robert Israel, Table of n, a(n) for n = 1..500

Cf. A000110, A005596, A350429.

filter:= proc(p) local b;
  b:= combinat:-bell(p-1);
  numtheory:-order(b,p) = p-1
end proc:
select(filter, [seq(ithprime(i),i=1..200)]); # Robert Israel, May 04 2023

Corrected by Robert Israel, May 04 2023

cp's OEIS Frontend

A120629 Numbers k with property that -k is not a perfect power and the squarefree part of -k is not congruent to 1 modulo 4.

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A146481 Decimal expansion of Product_{n>=2} (1 - 1/(n*(n-1))).

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A146482 Decimal expansion of Product_{q in A001358} (1-1/(q*(q-1))).

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A146483 Decimal expansion of Product_{q in A014612} (1-1/(q*(q-1))).

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A146484 Decimal expansion of Product_{q in A014613} (1-1/(q*(q-1))).

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A271869 Decimal expansion of Matthews' constant C_3, an analog of Artin's constant for primitive roots.

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A306198 Multiplicative with a(p^e) = p^(e-1)*(p^2 - p - 1).

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A333315 a(n) = Sum_{k=1..n} phi(prime(k)-1), where phi is the Euler totient function (A000005).

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