A005596 -id:A005596 - cp's OEIS frontend

A039787 Primes p such that p-1 is squarefree.

2, 3, 7, 11, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107, 131, 139, 167, 179, 191, 211, 223, 227, 239, 263, 283, 311, 331, 347, 359, 367, 383, 419, 431, 439, 443, 463, 467, 479, 499, 503, 547, 563, 571, 587, 599, 607, 619, 643, 647, 659, 683, 691, 719, 743

Offset: 1

Olivier Gérard

nonn

An equivalent definition: numbers n such that phi(n) is equal to the squarefree kernel of n-1.
Minimal value of first differences (between odd terms) is 4. - Zak Seidov, Apr 16 2013
The density of this set in A000040 is Artin's constant A = A005596 = 37.39...%, see Mirsky. - Charles R Greathouse IV, Oct 26 2015

			phi(43)=42, 42=2^1*3^1*7^1, 2*3*7=42.
p=223 is here because p-1=222=2*3*37

N. J. A. Sloane, Table of n, a(n) for n = 1..25000, Oct 25 2015 (extending earlier b-file of _Zak Seidov_)
Theodor Estermann, Einige Sätze über quadratfreie Zahlen, Math. Ann. 105:1 (1931), pp. 653-662.
Leon Mirsky, The number of representations of an integer as the sum of a prime and a k-free integer, American Mathematial Monthly 56:1 (1949), pp. 17-19.

Cf. A000010, A007947, A049092 (complement).

[p: p in PrimesUpTo(780) | IsSquarefree(p-1)];  // Bruno Berselli, Mar 03 2011

isA039787 := proc(n)
    if isprime(n) then
        numtheory[issqrfree](n-1) ;
    else
        false;
    end if;
end proc:
for n from 2 to 100 do
    if isA039787(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Apr 17 2013
with(numtheory): lis:=[]; for n from 1 to 10000 do if issqrfree(ithprime(n)-1) then lis:=[op(lis), ithprime(n)]; fi; od: lis; # N. J. A. Sloane, Oct 25 2015

Select[Prime[Range[132]],SquareFreeQ[#-1]&](* Zak Seidov, Aug 22 2012 *)

is(n)=isprime(n) && issquarefree(n-1) \\ Charles R Greathouse IV, Jul 02 2013

forprime(p=2, 1e3, if(issquarefree(p-1), print1(p", "))); \\ Altug Alkan, Oct 26 2015

More terms from Labos Elemer

A001609 a(1) = a(2) = 1, a(3) = 4; thereafter a(n) = a(n-1) + a(n-3).

Original entry on oeis.org

1, 1, 4, 5, 6, 10, 15, 21, 31, 46, 67, 98, 144, 211, 309, 453, 664, 973, 1426, 2090, 3063, 4489, 6579, 9642, 14131, 20710, 30352, 44483, 65193, 95545, 140028, 205221, 300766, 440794, 646015, 946781, 1387575, 2033590, 2980371, 4367946, 6401536, 9381907

Offset: 1

N. J. A. Sloane

This comment covers a family of sequences which satisfy a recurrence of the form a(n) = a(n-1) + a(n-m), with a(n) = 1 for n = 1...m-1, a(m) = m + 1. The generating function is (x + m*x^m)/(1 - x - x^m). Also a(n) = 1 + n*Sum_{i=1..n/m} binomial(n-1-(m-1)*i, i-1)/i. This gives the number of ways to cover (without overlapping) a ring lattice (or necklace) of n sites with molecules that are m sites wide. Special cases: m=2: A000204, m=3: A001609, m=4: A014097, m=5: A058368, m=6: A058367, m=7: A058366, m=8: A058365, m=9: A058364.
The sequence defined by {a(n) - 1} plays a role for the computation of A065414, A146486, A146487, and A146488 equivalent to the role of A001610 for A005596, A146482, A146483 and A146484, see the variable a_{2,n} in arXiv:0903.2514. - R. J. Mathar, Mar 28 2009
Except for n = 2, a(n) is the number of digits in n-th term of A049064. This can be derived form the T. Sillke link below. - Jianing Song, Apr 28 2019

			G.f. = x + x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 10*x^6 + 15*x^7 + 21*x^8 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Indranil Ghosh, Table of n, a(n) for n = 1..6012 (terms 1..500 from T. D. Noe)
Ignacio Cascudo, On squares of cyclic codes, arXiv:1703.01267 [cs.IT], 2017. See Theorem 6.1/Table 1.
Johann Cigler, Some remarks on generalized Fibonacci and Lucas polynomials, arXiv:1912.06651 [math.CO], 2019.
E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
Daniel C. Fielder, Special integer sequences controlled by three parameters, Fibonacci Quarterly 6, 1968, 64-70.
Daniel C. Fielder, Errata:Special integer sequences controlled by three parameters, Fibonacci Quarterly 6, 1968, 64-70.
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 91.
Matthew Macauley, Jon McCammond, Henning S. Mortveit, Dynamics groups of asynchronous cellular automata, Journal of Algebraic Combinatorics, Vol 33, No 1 (2011), pp. 11-35.
M. Newman and D. Shanks, On a sequence arising in series for pi, Math. Comp., 42 (1984), 199-217 (see Eq. 29).
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Souvik Roy, Nazim Fatès, and Sukanta Das, Reversibility of Elementary Cellular Automata with fully asynchronous updating: an analysis of the rules with partial recurrence, hal-04456320 [nlin.CG], [cs], 2024. See p. 19.
T. Sillke, The binary form of Conway's sequence
Z. Skupien, Sparse Hamiltonian 2-decompositions together with exact count of numerous Hamiltonian cycles, Discr. Math., 309 (2009), 6382-6390. - _N. J. A. Sloane_, Feb 12 2010
Index entries for linear recurrences with constant coefficients, signature (1,0,1).

Cf. A000204, A014097, A000079, A003269, A003520, A005708, A005709, A005710, A000930, A218439.

Cf. also A049064, A049194.

I:=[1,1,4]; [n le 3 select I[n] else Self(n-1)+Self(n-3): n in [1..45]]; // Vincenzo Librandi, Jun 28 2015

A001609:=-(1+3*z**2)/(-1+z+z**3); # Simon Plouffe in his 1992 dissertation
f:= gfun:-rectoproc({a(n) = a(n-1) + a(n-3), a(1)=1,a(2)=1,a(3)=4},a(n),remember):
map(f, [$1..100]); # Robert Israel, Jun 29 2015

Table[Tr[MatrixPower[{{0, 0, 1}, {1, 0, 0}, {0, 1, 1}}, n]], {n, 1, 60}] (* Artur Jasinski, Jan 10 2007 *)
Table[ HypergeometricPFQ[{1/3 - n/3, 2/3 - n/3, -(n/3)}, {1/2 - n/2, 1 - n/2}, -(27/4)], {n, 20}] (* Alexander R. Povolotsky, Nov 21 2008 *)
a[1] = a[2] = 1; a[3] = 4; m = 3; a[n_] := 1 + n*Sum [Binomial [n - 1 - (m - 1)*i, i - 1]/i, {i, n/m}] A001609 = Table[a[n], {n, 100}] (* Zak Seidov, Nov 21 2008 *)
LinearRecurrence[{1, 0, 1}, {1, 1, 4}, 50] (* Vincenzo Librandi, Jun 28 2015 *)

{a(n) = if( n<1, n=-n; polcoeff( (3 + x^2) / (1 + x^2 - x^3) + x * O(x^n), n), polcoeff( x * (1 + 3*x^2) / (1 - x - x^3) + x * O(x^n), n))}; /* Michael Somos, Aug 15 2016 */

G.f.: x*(1 + 3*x^2)/(1 - x - x^3).
a(n) = trace of successive powers of matrix ({{0,0,1},{1,0,0},{0,1,1}})^n. - Artur Jasinski, Jan 10 2007
a(n) = A000930(n) + 3*A000930(n-2). - R. J. Mathar, Nov 16 2007
Logarithmic derivative of Narayana's cows sequence A000930. - Paul D. Hanna, Oct 28 2012
a(n) = w1^n + w2^n + w3^n, where w1,w2,w3 are the roots of the cubic: (-1 - x^2 + x^3), see A092526. - Gerry Martens, Jun 27 2015

Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000
More terms from Michael Somos, Oct 03 2002
Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

A007351 Where prime race 4m-1 vs. 4m+1 is tied.

Original entry on oeis.org

2, 5, 17, 41, 461, 26833, 26849, 26863, 26881, 26893, 26921, 616769, 616793, 616829, 616843, 616871, 617027, 617257, 617363, 617387, 617411, 617447, 617467, 617473, 617509, 617531, 617579, 617681, 617707, 617719, 618437, 618521, 618593, 618637

Offset: 1

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

Primes p such that the number of primes <= p of the form 4m-1 is equal to the number of primes <= p of the form 4m+1.

Starting from a(27410)=9103362505753 the sequence includes the 8th sign-changing zone predicted by C. Bays et al. The sequence with the first 8 sign-changing zones contains 419467 terms (see a-file) with a(419467)=9543313015351 as its last term. - Sergei D. Shchebetov, Oct 15 2017

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrey S. Shchebetov and Sergei D. Shchebetov, Table of n, a(n) for n = 1..100000 (first 1000 terms from T. D. Noe)
A. Alahmadi, M. Planat, and P. Solé, Chebyshev's bias and generalized Riemann hypothesis, HAL Id: hal-00650320.
C. Bays and R. H. Hudson, Numerical and graphical description of all axis crossing regions for moduli 4 and 8 which occur before 10^12, International Journal of Mathematics and Mathematical Sciences, vol. 2, no. 1, pp. 111-119, 1979.
C. Bays, K. Ford, R. H. Hudson and M. Rubinstein, Zeros of Dirichlet L-functions near the real axis and Chebyshev's bias, J. Number Theory 87 (2001), pp.54-76.
M. Deléglise, P. Dusart, and X. Roblot, Counting Primes in Residue Classes, Mathematics of Computation, American Mathematical Society, 2004, 73 (247), pp.1565-1575.
A. Granville and G. Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
M. Rubinstein and P. Sarnak, Chebyshev’s bias, Experimental Mathematics, Volume 3, Issue 3, 1994, Pages 173-197.
Andrey S. Shchebetov and Sergei D. Shchebetov, First 419467 terms (zipped file)
Eric Weisstein's World of Mathematics, Prime Quadratic Effect.
Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993

Cf. A007350, A038691.

Cf. A156749 Sequence showing Chebyshev bias in prime races (mod 4). [From Daniel Forgues, Mar 26 2009]

Prime@ Position[Fold[Append[#1, #1[[-1]] + If[Mod[#2, 4] == 3, {1, 0}, {0, 1}]] &, {{0, 0}}, Prime@ Range[2, 10^5]], ?(SameQ @@ # &)][[All, 1]] (* _Michael De Vlieger, May 27 2018 *)

lista(nn) = {nb = 0; forprime(p=2, nn, m = (p % 4); if (m == 1, nb++, if (m == 3, nb--)); if (!nb, print1(p, ", ")););} \\ Michel Marcus, Oct 05 2017

from sympy import nextprime; a, p = 0, 2; R = [p]
while p < 618637:
    p=nextprime(p); a += p%4-2
    if a == 0: R.append(p)
print(*R, sep = ', ')  # Ya-Ping Lu, Jan 18 2025

Corrected and extended by Enoch Haga, Feb 24 2004

A272030 Decimal expansion of C = log(2*Pi) + B_3 (where B_3 is A083343), one of Euler totient constants.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 25 2016

			3.17045934214256636532648824888226302856125443631798948742143398...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.7 Euler totient constants, p. 117.

Cf. A001620, A005596, A065484, A065485, A080130, A082695, A082695, A083343.

digits = 98; B3 = EulerGamma - NSum[PrimeZetaP'[n], {n, 2, Infinity}, WorkingPrecision -> 2 digits, NSumTerms -> 200]; RealDigits[Log[2 Pi] + B3, 10, digits][[1]]

C = log(2*Pi) + EulerGamma - Sum_{n >= 2} P'(n), where P'(n) is the prime zeta P function derivative.

A007352 Where the prime race 3k-1 vs. 3k+1 changes leader.

Original entry on oeis.org

2, 608981813029, 608981813507, 608981813683, 608981813819, 608981814127, 608981814143, 608981818999, 608981820977, 608981826877, 608981826977, 608981827873, 608981828201, 608981836363, 608981836493, 608981836681, 608981836973, 608981836993, 608981837063

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v, Apr 28 1994

nonn

Terms a(2n+1) form a subsequence of A098044.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence, although the terms are incorrect - see A185703).

Donovan Johnson, Table of n, a(n) for n = 1..39215
C. Bays and R. H. Hudson, Details of the first region of integers x with pi_{3,2}(x) < pi_{3,1}(x), Math. Comp. 32 (1978), 571-576.
A. Granville and G. Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
G. Martin, Asymmetries in the Shanks-Renyi Prime number race, arXiv:math/0010086 [math.NT], 2000.
Pieter Moree, Chebyshev's bias for composite numbers with restricted prime divisors, Mathematics of computation 73.245 (2004): 425-449. See page 425.
M. Rubinstein and P. Sarnak, Chebyshev's Bias, Exper. Math. 3 (4) (1994) 209.
Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993

Cf. A007352, A007350, A007353, A007354, A098044, A297406, A297407, A297408, A297410, A297411.

Terms from a(3) onwards corrected by Max Alekseyev, Feb 10 2011

A049097 Primes p such that p+1 is squarefree.

Original entry on oeis.org

2, 5, 13, 29, 37, 41, 61, 73, 101, 109, 113, 137, 157, 173, 181, 193, 229, 257, 277, 281, 313, 317, 353, 373, 389, 397, 401, 409, 421, 433, 457, 461, 509, 541, 569, 601, 613, 617, 641, 653, 661, 673, 677, 709, 733, 757, 761, 769, 797, 821, 829, 853, 857

Offset: 1

Labos Elemer

nonn

Numbers k such that core(sigma(k)) = k + 1 where core(k) is the squarefree part of k (A007913). - Benoit Cloitre, May 01 2002

This sequence is infinite and its relative density in the sequence of primes is equal to Artin's constant (A005596): Product_{p prime} (1-1/(p*(p-1))) = 0.373955... (Mirsky, 1949). - Amiram Eldar, Dec 29 2020

			29 is included since 29 + 1 = 30 = 2*3*5 is squarefree.
17 is not here because 18 is divisible by a square, 9.

Robert Israel, Table of n, a(n) for n = 1..10000
Leon Mirsky, The number of representations of an integer as the sum of a prime and a k-free integer, The American Mathematical Monthly, Vol. 56, No. 1 (1949), pp. 17-19.

Cf. A000040, A005117, A039787.

Cf. A000203, A005596, A007913, A077067, A160696.

[ p: p in PrimesUpTo(900) | IsSquarefree(p+1) ]; // Vincenzo Librandi, Dec 25 2010

N:= 10000; # to get all entries up to N
A049097:= select(t -> isprime(t) and numtheory:-issqrfree(t+1), [2, seq(1+2*k,k=1..floor((N-1)/2))]); # Robert Israel, May 11 2014

Select[Prime[Range[100]], SquareFreeQ[# + 1] &] (* Zak Seidov, Feb 08 2016 *)

lista(nn) = forprime(p=1, nn, if (issquarefree(p+1), print1(p, ", "))); \\ Michel Marcus, Jan 08 2015

A160696(a(n)) = 1. - Reinhard Zumkeller, May 24 2009

a(n) = A077067(n)-1. - Zak Seidov, Mar 19 2016

A065414 Decimal expansion of rank 2 Artin constant Product_{p prime} (1-1/(p^3-p^2)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 15 2001

			0.697501358496365903284670350820922924...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.4, p. 105.

R. J. Mathar, Hardy-Littlewood constants embedded into infinite products.., arxiv:0903.2514 [math.NT], 2009-2011, variable A_1^(2).
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
Index entries for sequences related to Artin's conjecture.

Cf. A005596, A065417.

digits = 99; m0 = 1000; dm = 100; Clear[s]; r[n_] := RootSum[-1 - #^2 + #^3 &, #^n&] - 1; s[m_] := s[m] = NSum[-r[n] PrimeZetaP[n]/n, {n, 3, m}, NSumTerms -> m0, WorkingPrecision -> 300] // 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]; RealDigits[s[m], 10, digits][[1]] (* Jean-François Alcover, Apr 14 2016 *)

prodeulerrat(1-1/(p^3-p^2)) \\ Amiram Eldar, Mar 12 2021

A090866 Primes p == 1 (mod 4) such that (p-1)/4 is prime.

Original entry on oeis.org

13, 29, 53, 149, 173, 269, 293, 317, 389, 509, 557, 653, 773, 797, 1109, 1229, 1493, 1637, 1733, 1949, 1997, 2309, 2477, 2693, 2837, 2909, 2957, 3413, 3533, 3677, 3989, 4133, 4157, 4253, 4349, 4373, 4493, 4517, 5189, 5309, 5693, 5717, 5813, 6173, 6197

Offset: 1

Benoit Cloitre, Feb 12 2004

nonn

Same as Chebyshev's subsequence of the primes with primitive root 2, because Chebyshev showed that 2 is a primitive root of all primes p = 4*q+1 with q prime. If the sequence is infinite, then Artin's conjecture ("every nonsquare positive integer n is a primitive root of infinitely many primes q") is true for n = 2. - Jonathan Sondow, Feb 04 2013

Albert H. Beiler: Recreations in the theory of numbers. New York: Dover, (2nd ed.) 1966, p. 102, nr. 5.
P. L. Chebyshev, Theory of congruences. Elements of number theory, Chelsea, 1972, p. 306.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Artin's conjecture

Cf. A001122, A005385, A005596, A023212, A221981, A222008.

f:=[n: n in [1..2000] | IsPrime(n) and IsPrime(4*n+1)]; [4*f[n] + 1: n in [1..50]]; // G. C. Greubel, Feb 08 2019

Select[Prime[Range[1000]], Mod[#, 4]==1 && PrimeQ[(#-1)/4] &] (* G. C. Greubel, Feb 08 2019 *)

isok(p) = isprime(p) && !frac(q=(p-1)/4) && isprime(q); \\ Michel Marcus, Feb 09 2019

a(n) = 4*A023212(n) + 1.

A003147 Primes p with a Fibonacci primitive root g, i.e., such that g^2 = g + 1 (mod p).

Original entry on oeis.org

5, 11, 19, 31, 41, 59, 61, 71, 79, 109, 131, 149, 179, 191, 239, 241, 251, 269, 271, 311, 359, 379, 389, 409, 419, 431, 439, 449, 479, 491, 499, 569, 571, 599, 601, 631, 641, 659, 701, 719, 739, 751, 821, 839, 929, 971, 1019, 1039, 1051, 1091, 1129, 1171, 1181, 1201, 1259, 1301

Offset: 1

N. J. A. Sloane

Primes p with a primitive root g such that g^2 = g + 1 (mod p).
Not the same as primes with a Fibonacci number as primitive root; cf. A083701. - Jonathan Sondow, Feb 17 2013
For all except the initial term 5, these are numbers such that the Pisano period equals 1 less than the Pisano number, i.e., where A001175(n) = n-1. - Matthew Goers, Sep 20 2013
As shown in the paper by Brison, these are also the primes p such that there is a Fibonacci-type sequence (mod p) that begins with (1,b) and encounters all numbers less than p in the first p-1 iterations (for some b). - T. D. Noe, Feb 26 2014
Shanks (1972) conjectured that the relative asymptotic density of this sequence in the sequence of primes is 27*c/38 = 0.2657054465..., where c is Artin's constant (A005596). The conjecture was proved on the assumption of a generalized Riemann hypothesis by Lenstra (1977) and Sander (1990). - Amiram Eldar, Jan 22 2022

			3 is a primitive root mod 5, and 3^2 = 3 + 1 mod 5, so 5 is a member. - _Jonathan Sondow_, Feb 17 2013

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from Noe)
Bob Bastasz, Lyndon words of a second-order recurrence, Fibonacci Quarterly, Vol. 58, No. 5 (2020), pp. 25-29.
Owen J. Brison, Complete Fibonacci sequences in finite fields, Fibonacci Quarterly, Vol. 30, No. 4 (1992), pp. 295-304.
Alexandru Gica, Quadratic Residues in Fibonacci Sequences, Fibonacci Quart., Vol. 46/47, No. 1 (2008/2009), pp. 68-72. See Theorem 5.1.
Liang-Chung Hsia, Hua-Chieh Li, and Wei-Liang Sun, Certain Diagonal Equations and Conflict-Avoiding Codes of Prime Lengths, arXiv:2302.00920 [math.NT], 2023.
H. W. Lenstra, Jr., On Artin's conjecture and Euclid's algorithm in global fields, Invent. Math., Vol. 42 (1977), pp. 202-224; alternative link.
J. W. Sander, On Fibonacci primitive roots, Fibonacci Quarterly, Vol. 28, No. 1 (1990), pp. 79-80.
Daniel Shanks, Fibonacci primitive roots, end of article, Fibonacci Quarterly, Vol. 10, No. 2 (1972), pp. 163-168, 181.
Daniel Shanks and Larry Taylor, An Observation of Fibonacci Primitive Roots, Fibonacci Quarterly, Vol. 11, No. 2 (1973), pp. 159-160.
Index entries for primes by primitive root.

Subsequence of A038872.
Cf. A001175, A005596, A083701.
See also A106535.

filter:=proc(n) local g,r;
if not isprime(n) then return false fi;
r:= [msolve(g^2 -g - 1, n)][1];
numtheory:-order(rhs(op(r)),n) = n-1
end proc:
select(filter, [5,seq(seq(10*i+j,j=[1,9]),i=1..1000)]); # Robert Israel, May 22 2015

okQ[p_] := AnyTrue[PrimitiveRootList[p], Mod[#^2, p] == Mod[#+1, p]&]; Select[Prime[Range[300]], okQ] (* Jean-François Alcover, Jan 04 2016 *)

is(n)=if(kronecker(5,n)<1||!isprime(n), return(n==5)); my(s=sqrt(Mod(5,n))); znorder((1+s)/2)==n-1 || znorder((1-s)/2)==n-1 \\ Charles R Greathouse IV, May 22 2015

More terms from David W. Wilson
Cross-reference from Charles R Greathouse IV, Nov 05 2009
Definition clarified by M. F. Hasler, Jun 05 2018

A003968 Möbius transform of A003959.

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 7, 18, 12, 10, 11, 18, 13, 14, 15, 54, 17, 24, 19, 30, 21, 22, 23, 54, 30, 26, 48, 42, 29, 30, 31, 162, 33, 34, 35, 72, 37, 38, 39, 90, 41, 42, 43, 66, 60, 46, 47, 162, 56, 60, 51, 78, 53, 96, 55, 126, 57, 58, 59, 90, 61, 62, 84, 486, 65, 66, 67, 102, 69

Offset: 1

Marc LeBrun

a(n) = n for squarefree n; otherwise, a(n) > n. - Ivan Neretin, May 13 2015

Dirichlet inverse of A062953. - Werner Schulte, Oct 25 2018

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A003959, A005596, A062953, A104141.

Table[{pp, aa} = Transpose[FactorInteger[n]]; Times @@ (pp*(pp + 1)^(aa - 1)), {n, 70}]  (* Ivan Neretin, May 13 2015 *)

a(n) = {my(f=factor(n)); for (i=1, #f~, p= f[i, 1]; f[i, 1] = p*(p+1)^(f[i,2]-1); f[i, 2] = 1); factorback(f);} \\ Michel Marcus, Feb 26 2015

Multiplicative with a(p^e) = p(p+1)^(e-1). - David W. Wilson, Sep 01 2001

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 + 1/(p^3 - p^2 - p)) = A104141/A005596 = 0.8128327996... . - Amiram Eldar, Oct 23 2022

More terms from David W. Wilson, Aug 29 2001

cp's OEIS Frontend

A039787 Primes p such that p-1 is squarefree.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Extensions

A001609 a(1) = a(2) = 1, a(3) = 4; thereafter a(n) = a(n-1) + a(n-3).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A007351 Where prime race 4m-1 vs. 4m+1 is tied.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A272030 Decimal expansion of C = log(2*Pi) + B_3 (where B_3 is A083343), one of Euler totient constants.

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Mathematica

Formula

A007352 Where the prime race 3k-1 vs. 3k+1 changes leader.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A049097 Primes p such that p+1 is squarefree.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A065414 Decimal expansion of rank 2 Artin constant Product_{p prime} (1-1/(p^3-p^2)).

Views