A005596 -id:A005596 - cp's OEIS frontend

A348030 a(n) = A003968(n) - n, where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).

0, 0, 0, 2, 0, 0, 0, 10, 3, 0, 0, 6, 0, 0, 0, 38, 0, 6, 0, 10, 0, 0, 0, 30, 5, 0, 21, 14, 0, 0, 0, 130, 0, 0, 0, 36, 0, 0, 0, 50, 0, 0, 0, 22, 15, 0, 0, 114, 7, 10, 0, 26, 0, 42, 0, 70, 0, 0, 0, 30, 0, 0, 21, 422, 0, 0, 0, 34, 0, 0, 0, 144, 0, 0, 15, 38, 0, 0, 0, 190, 111, 0, 0, 42, 0, 0, 0, 110, 0, 30, 0, 46

Offset: 1

Antti Karttunen, Oct 19 2021

nonn

Möbius transform of A348029(n), which is A003959(n) - sigma(n).

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A003959, A003968, A005117 (positions of zeros), A005596, A008683, A104141, A348029, A348036.

f[p_, e_] := p*(p + 1)^(e - 1); a[n_] := Times @@ f @@@ FactorInteger[n] - n; Array[a, 100] (* Amiram Eldar, Oct 20 2021 *)

A003968(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); }
A348030(n) = (A003968(n)-n);

a(n) = A003968(n) - n.
a(n) = Sum_{d|n} A008683(n/d) * A348029(d).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p^3 - p^2 - p)) - 1 = A104141/A005596 - 1 = 0.625665... . - Amiram Eldar, May 29 2025

A119534 Largest prime divisor of numerator of the n-th Artin's product.

Original entry on oeis.org

5, 19, 41, 109, 109, 271, 271, 271, 811, 929, 929, 929, 929, 2161, 2161, 2161, 3659, 4421, 4969, 4969, 4969, 4969, 4969, 9311, 10099, 10099, 10099, 10099, 10099, 16001, 17029, 17029, 19181, 22051, 22051, 22051, 22051, 22051, 22051, 22051, 32579

Offset: 2

Alexander Adamchuk, Jul 27 2006

Artin's constant (A005596) is equal to Product[1-1/(Prime[k]*(Prime[k]-1)),{k,1,Infinity}]. n-th Artin's product is Product[1-1/(Prime[k]*(Prime[k]-1)),{k,1,n}]. a(n) is prime from A091568 of the form p^2-p-1, where p is prime from A091567.

Amiram Eldar, Table of n, a(n) for n = 2..1000
Eric Weisstein's World of Mathematics, Artin's Constant.

Cf. A091568, A091567, A005596, A048296.

[Max(PrimeDivisors(Numerator(&*[1-1/(NthPrime(k)^2-NthPrime(k)):k in [1..n]]))): n in [2..45]]; // Marius A. Burtea, Feb 18 2020

Table[Max[FactorInteger[Numerator[Product[1-1/(Prime[k]*(Prime[k]-1)),{k,1,n}]]]],{n,2,100}]

a(n) = Max[FactorInteger[Numerator[Product[1-1/(Prime[k]*(Prime[k]-1)),{k,1,n}]]]].

A192864 Lower 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, 17, 23, 29, 31, 41, 43, 47, 53, 59, 61, 67, 71, 79, 83, 89, 97, 103, 107, 113, 131, 137, 139, 149, 157, 167, 173, 179, 191, 193, 211, 223, 227, 229, 233, 239, 241, 257, 263, 269, 277, 281, 283, 293, 311, 313, 317, 331, 337, 347, 349, 353, 359

Offset: 1

Charles R Greathouse IV, Jul 11 2011

nonn

Broughan & Qizhi show that this sequence has relative density 2*A 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.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Kevin A. Broughan and Zhou Qizhi, Flat primes and thin primes, Bulletin of the Australian Mathematical Society, Vol. 82, No. 2 (2010), pp. 282-292, alternative link.

Subsequence of A192863.

Cf. A192861, A192862, A005596.

Select[Range[3, 360, 2], PrimeQ[#] && SquareFreeQ[(# - 1)/2^IntegerExponent[# - 1, 2]] &] (* Amiram Eldar, Aug 30 2020 *)

is(n)=n%2&&isprime(n)&&issquarefree((n-1)>>valuation(n-1,2)) \\ corrected by Amiram Eldar, Aug 30 2020

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

Data corrected by Amiram Eldar, Aug 30 2020

A323576 Primes p such that 2 is a primitive root modulo p while 128 is not.

Original entry on oeis.org

29, 197, 211, 379, 421, 491, 547, 659, 701, 757, 827, 883, 1373, 1499, 1667, 1877, 2213, 2269, 2339, 2437, 2549, 2843, 3011, 3067, 3347, 3557, 3571, 3613, 3851, 3907, 4019, 4229, 4243, 4397, 4621, 4691, 4789, 4957, 5573, 5741, 5923, 6203, 6469, 6637, 6763, 6917

Offset: 1

Jianing Song, Aug 30 2019

nonn

Primes p such that 2 is a primitive root modulo p (i.e., p is in A001122) and that p == 1 (mod 7).

According to Artin's conjecture, the number of terms <= N is roughly ((6/41)*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.

Cf. A001122, A005596, A000720.

Primes p such that 2 is a primitive root modulo p and that p == 1 (mod q): A307627 (q=3), A307628 (q=5), this sequence (q=7), A323577 (q=11), A323590 (q=13).

forprime(p=3, 7000, if(znorder(Mod(2, p))==(p-1) && p%7==1, print1(p, ", ")))

A323577 Primes p such that 2 is a primitive root modulo p while 2048 is not.

Original entry on oeis.org

67, 419, 661, 859, 947, 1123, 1277, 1453, 2069, 2267, 2333, 2531, 2707, 2861, 3037, 3323, 3499, 3851, 3917, 4093, 4357, 4621, 4973, 5171, 5501, 6029, 6469, 6491, 6733, 7019, 7283, 7349, 7459, 7547, 7789, 7877, 8053, 8669, 8867, 8933, 9901, 9923, 10099, 10253, 10891, 10979

Offset: 1

Jianing Song, Aug 30 2019

nonn

Primes p such that 2 is a primitive root modulo p (i.e., p is in A001122) and that p == 1 (mod 11).

According to Artin's conjecture, the number of terms <= N is roughly ((10/109)*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.

Cf. A001122, A005596, A000720.

Primes p such that 2 is a primitive root modulo p and that p == 1 (mod q): A307627 (q=3), A307628 (q=5), A323576 (q=7), this sequence (q=11), A323590 (q=13).

forprime(p=3, 12000, if(znorder(Mod(2, p))==(p-1) && p%11==1, print1(p, ", ")))

A323590 Primes p such that 2 is a primitive root modulo p while 8192 is not.

Original entry on oeis.org

53, 131, 443, 547, 677, 859, 1171, 1301, 1483, 2029, 2237, 2549, 2861, 2939, 3797, 4603, 5227, 5851, 6397, 6709, 6917, 7229, 7307, 7411, 7541, 7853, 8243, 8269, 8867, 8971, 9283, 9491, 9803, 9907, 10037, 10141, 10427, 10973, 11779, 11909, 11987, 12611, 12637, 12923

Offset: 1

Jianing Song, Aug 30 2019

nonn

Primes p such that 2 is a primitive root modulo p (i.e., p is in A001122) and that p == 1 (mod 13).

According to Artin's conjecture, the number of terms <= N is roughly ((12/155)*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).

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

Cf. A001122, A005596, A000720.

Primes p such that 2 is a primitive root modulo p and that p == 1 (mod q): A307627 (q=3), A307628 (q=5), A323576 (q=7), A323577 (q=11), this sequence (q=13).

filter:= proc(p) isprime(p) and numtheory:-order(2,p) = p-1 end proc:
select(filter, [seq(i, i = 1 .. 13000, 26)]); # Robert Israel, Dec 20 2023

forprime(p=3, 13000, if(znorder(Mod(2, p))==(p-1) && p%13==1, print1(p, ", ")))

A323617 Primes p such that 3 is a primitive root modulo p while 243 is not.

Original entry on oeis.org

31, 101, 211, 281, 331, 401, 461, 521, 571, 631, 641, 691, 701, 751, 811, 821, 881, 941, 1061, 1231, 1291, 1301, 1361, 1481, 1601, 1721, 1831, 1901, 1951, 2011, 2081, 2141, 2311, 2371, 2381, 2731, 2741, 2801, 2861, 3041, 3271, 3331, 3391, 3461, 3571, 3581, 3701, 3761, 3821, 3931

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 5).

According to 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).

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.

Cf. A019334, A005596, A000720.

Primes p such that 3 is a primitive root modulo p and that p == 1 (mod q): A323594 (q=3), this sequence (q=5), A323628 (q=7).

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

A323628 Primes p such that 3 is a primitive root modulo p while 2187 is not.

Original entry on oeis.org

29, 43, 113, 127, 197, 211, 281, 379, 449, 463, 617, 631, 701, 953, 1373, 1709, 1723, 2129, 2143, 2213, 2311, 2381, 2549, 2633, 2647, 2731, 2801, 2969, 3137, 3389, 3557, 3571, 3823, 4159, 4229, 4243, 4327, 4397, 4481, 4649, 4663, 4817, 4831, 4999, 5237, 5419

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 7).

According to Artin's conjecture, the number of terms <= N is roughly ((6/41)*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).

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

Cf. A019334, A005596, A000720.

Primes p such that 3 is a primitive root modulo p and that p == 1 (mod q): A323594 (q=3), A323617 (q=5), this sequence (q=7).

select(p -> isprime(p) and numtheory:-order(3,p)=p-1, [seq(i,i=1..10000,7)]); # Robert Israel, Sep 01 2019

forprime(p=5, 5500, if(znorder(Mod(3, p))==(p-1) && p%7==1, print1(p, ", ")))

A336654 Numbers k such that lambda(k) is squarefree, where lambda is the Carmichael lambda function (A002322).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 18, 21, 22, 23, 24, 28, 31, 33, 36, 42, 43, 44, 46, 47, 49, 56, 59, 62, 63, 66, 67, 69, 71, 72, 77, 79, 83, 84, 86, 88, 92, 93, 94, 98, 99, 103, 107, 118, 121, 124, 126, 129, 131, 132, 134, 138, 139, 141, 142, 147, 154, 158, 161

Offset: 1

Amiram Eldar, Jul 28 2020

nonn

			6 is a term since lambda(6) = 2 is squarefree.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Imre Kátai, Square-free values of the Carmichael function, Mathematica Pannonica, Vol. 16, No. 2 (2005), pp. 199-203.
Francesco Pappalardi, Filip Saidak and Igor E. Shparlinski, Square-free values of the Carmichael function, Journal of Number Theory, Vol. 103, No. 1 (2003), pp. 122-131.

Cf. A002322, A005117, A005596, A049149, A336655, A336656.

Select[Range[160], SquareFreeQ[CarmichaelLambda[#]] &]

The number of terms not exceeding x is (k + o(1)) * x/(log(x)^(1-a)), where a = 0.373955... is Artin's constant (A005596), and k = 0.80328... is another constant (Pappalardi et al., 2003).

A007349 Primes with both 10 and -10 as primitive root.

Original entry on oeis.org

17, 29, 61, 97, 109, 113, 149, 181, 193, 229, 233, 257, 269, 313, 337, 389, 433, 461, 509, 541, 577, 593, 701, 709, 821, 857, 937, 941, 953, 977, 1021, 1033, 1069, 1097, 1109, 1153, 1181, 1193, 1217, 1229, 1297, 1301, 1381, 1429, 1433, 1549, 1553, 1621, 1697, 1709, 1741, 1777, 1789, 1861, 1873, 1913, 1949

Offset: 1

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

nonn

Intersection of A006883 and A002144. - Davide Rotondo, May 21 2025

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 864.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993
Index entries for primes by primitive root

Cf. A002144, A006883.

pr=10; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == MultiplicativeOrder[-pr, # ] == #-1 &]
Select[Prime[Range[5,200]],PrimitiveRoot[#,10]==10&&PrimitiveRoot[#,#-10] == #-10&] (* Harvey P. Dale, Oct 10 2019 *)

forprime(p=11,2000,if(znorder(Mod(10,p))==p-1&&znorder(Mod(-10,p))==p-1,print1(p,", "))); \\ Joerg Arndt, May 21 2025

cp's OEIS Frontend

A348030 a(n) = A003968(n) - n, where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).

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A119534 Largest prime divisor of numerator of the n-th Artin's product.

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Magma

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A192864 Lower flat primes: odd primes p such that p-1 is a squarefree number times a power of two.

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A323576 Primes p such that 2 is a primitive root modulo p while 128 is not.

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A323577 Primes p such that 2 is a primitive root modulo p while 2048 is not.

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A323590 Primes p such that 2 is a primitive root modulo p while 8192 is not.

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Maple

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A323617 Primes p such that 3 is a primitive root modulo p while 243 is not.

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A323628 Primes p such that 3 is a primitive root modulo p while 2187 is not.

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