A005596 -id:A005596 - cp's OEIS frontend

A049098 Primes p such that p+1 is divisible by a square.

3, 7, 11, 17, 19, 23, 31, 43, 47, 53, 59, 67, 71, 79, 83, 89, 97, 103, 107, 127, 131, 139, 149, 151, 163, 167, 179, 191, 197, 199, 211, 223, 227, 233, 239, 241, 251, 263, 269, 271, 283, 293, 307, 311, 331, 337, 347, 349, 359, 367, 379, 383, 419, 431, 439, 443

Offset: 1

Labos Elemer

Numbers m such that A010051(m)*(1-A008966(m+1)) = 1. - Reinhard Zumkeller, May 21 2009

This sequence is infinite and its relative density in the sequence of primes is equal to 1 - Product_{p prime} (1-1/(p*(p-1))) = 1 - A005596 = 0.626044... (Mirsky, 1949). - Amiram Eldar, Feb 14 2021

			31 is a term because 32 is divisible by a square, 16.
101 is not a term because 102 = 2*3*17 is squarefree.

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, The American Mathematical Monthly, Vol. 56, No. 1 (1949), pp. 17-19.

Cf. A005596, A008966, A010051, A049097 (complement with respect to A000040), A160696.

a049098 n = a049098_list !! (n-1)
a049098_list = filter ((== 0) . a008966 . (+ 1)) a000040_list
-- Reinhard Zumkeller, Oct 18 2011

with(numtheory): a := proc (n) if isprime(n) = true and issqrfree(n+1) = false then n else end if end proc: seq(a(n), n = 1 .. 500); # Emeric Deutsch, Jun 21 2009

Select[Prime[Range[200]],!SquareFreeQ[#+1]&]   (* Harvey P. Dale, Mar 27 2011 *)
Select[Prime[Range[200]], MoebiusMu[# + 1] == 0 &] (* Alonso del Arte, Oct 18 2011 *)

forprime(p=2,1e4,if(!issquarefree(p+1),print1(p", "))) \\ Charles R Greathouse IV, Oct 18 2011

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

A065415 Decimal expansion of Product_{p prime} (1-1/(p^4-p^3)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 15 2001

			0.85654044485354217442616798413595388...

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 over all positive integers, arXiv:0903.2514 [math.NT], 2009-2011, constant A_1^(3) table 3.
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]

Cf. A005596, A065414, A065416.

digits = 99; $MaxExtraPrecision = 400; m0 = 1000; dm = 100; Clear[s]; LR = LinearRecurrence[{2, -1, 0, 1, -1}, {0, 0, 0, 4, 5, 6}, 2 m0]; r[n_Integer] := LR[[n]]; s[m_] := s[m] = NSum[-r[n] PrimeZetaP[n]/n, {n, 3, m}, NSumTerms -> m0, WorkingPrecision -> 400] // 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 15 2016 *)

prodeulerrat(1-1/(p^4-p^3)) \\ Amiram Eldar, Mar 13 2021

A065416 Decimal expansion of Product_{p prime} (1-1/(p^5-p^4)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 15 2001

			0.93126518416000433438923720555067698...

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

G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]

Cf. A005596, A065414, A065415.

digits = 99; $MaxExtraPrecision = 400; m0 = 1000; dm = 100; Clear[s]; LR = LinearRecurrence[{2, -1, 0, 0, 1, -1}, {0, 0, 0, 0, 5, 6}, 2 m0]; r[n_Integer] := LR[[n]]; s[m_] := s[m] = NSum[-r[n] PrimeZetaP[n]/n, {n, 5, m}, NSumTerms -> m0, WorkingPrecision -> 400] // 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 15 2016 *)

prodeulerrat(1-1/(p^5-p^4)) \\ Amiram Eldar, Mar 12 2021

A065488 Decimal expansion of Product_{p prime} (1 + 1/(p^2-p-1)).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Nov 19 2001

This is 1/Artin's constant, see A005596.

			2.67411272557002150896041183...

G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]

Cf. A005596, A048296, A351219.

$MaxExtraPrecision = 1200; digits = 99; terms = 1200; P[n_] := PrimeZetaP[n ]; LR = Join[{0, 0}, LinearRecurrence[{2, 0, -1}, {2, 3, 6}, term+10]]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*P[n - 1]/(n - 1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits+10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 18 2016 *)

prodeulerrat(1 + 1/(p^2-p-1)) \\ Amiram Eldar, Mar 15 2021

A019338 Primes with primitive root 8.

Original entry on oeis.org

3, 5, 11, 29, 53, 59, 83, 101, 107, 131, 149, 173, 179, 197, 227, 269, 293, 317, 347, 389, 419, 443, 461, 467, 491, 509, 557, 563, 587, 653, 659, 677, 701, 773, 797, 821, 827, 941, 947, 1019, 1061, 1091, 1109, 1187, 1229, 1259, 1277, 1283, 1301, 1307, 1373, 1427

Offset: 1

David W. Wilson

nonn

To allow primes less than the specified primitive root m (here, 8) 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 03 2019
Members of A001122 that are not congruent to 1 mod 3. - Robert Israel, Aug 12 2014
Terms greater than 3 are congruent to 5 or 11 modulo 24. - Jianing Song, May 12 2024 [Corrected on May 13 2025]

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for primes by primitive root

select(t -> isprime(t) and numtheory:-order(8,t) = t-1, [2*i+1 $ i=1..1000]); # Robert Israel, Aug 12 2014

pr=8; Select[Prime[Range[200]], 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[#, 3],20]]==1 &]+1
(* Gerry Martens, Apr 28 2015 *)
Join[{3,5},Select[Prime[Range[250]],PrimitiveRoot[#,8]==8&]] (* Harvey P. Dale, Aug 10 2019 *)

is(n)=isprime(n) && n>2 && 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,3)==1. - Gerry Martens, May 15 2015

On Artin's conjecture, a(n) ~ (5/3A) n log n, where A = A005596 is Artin's constant. - Charles R Greathouse IV, May 21 2015

A048296 Continued fraction for Artin's constant.

Original entry on oeis.org

0, 2, 1, 2, 14, 1, 1, 2, 3, 5, 1, 3, 1, 5, 1, 1, 2, 3, 5, 46, 2, 2, 4, 4, 2, 1, 6, 1, 1, 4, 2, 2, 1, 109, 1, 1, 4, 9, 3, 45, 8, 4, 1, 2, 1, 13, 13, 1, 1, 2, 1, 1, 2, 1, 4, 2, 3, 1, 17, 1, 1, 1, 6, 42, 1, 3, 1, 1, 4, 1, 1, 1, 1, 1, 2, 4, 5, 4, 1, 26, 1, 1, 74, 1, 1, 2, 1, 2, 2, 1, 1, 10, 1

Offset: 0

Fred Lunnon and Simon Plouffe, Dec 11 1999

			artin = 0.37395581361920228805... = 0 + 1/(2 + 1/(1 + 1/(2 + 1/(14 + ...)))). - _Harry J. Smith_, Apr 23 2009

See A005596 for further references.

Harry J. Smith, Table of n, a(n) for n = 0..995
Index entries for sequences related to Artin's conjecture

Cf. A005596.

digits = 105; m0 = 1000; dm = 100; Clear[s]; r[n_] := -1 + Fibonacci[n-1] + Fibonacci[n+1]; s[m_] := s[m] = NSum[-r[n] PrimeZetaP[n]/n, {n, 2, m}, NSumTerms -> m0, WorkingPrecision -> 400] // 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]; A = s[m]; ContinuedFraction[A, 93] (* Jean-François Alcover, Apr 15 2016 *)

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

A049092 Primes p such that p-1 is not squarefree.

Original entry on oeis.org

5, 13, 17, 19, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 127, 137, 149, 151, 157, 163, 173, 181, 193, 197, 199, 229, 233, 241, 251, 257, 269, 271, 277, 281, 293, 307, 313, 317, 337, 349, 353, 373, 379, 389, 397, 401, 409, 421, 433, 449, 457, 461, 487

Offset: 1

Labos Elemer

nonn

Primes p with mu(p-1)=0, where mu is the Möbius function. - T. D. Noe, Nov 03 2003
Primes p such that the sum of the primitive roots of p (see A088144) is 0 mod p. - Jon Wharf, Mar 12 2015
The relative density of this sequence within the primes is 1 - A005596 = 0.626044... - Amiram Eldar, Feb 10 2021

			p = 257 is here because p-1 = 256 = 2^8.
p = 997 is here because p-1 = 996 = 3*(2^2)*83.

T. D. Noe, Table of n, a(n) for n=1..1000
Eric Weisstein's World of Mathematics, Moebius Function.

Cf. A005596, A039787, A078330 (primes p with mu(p-1)=-1), A088179 (primes p such that mu(p-1)=1), A089451 (mu(p-1) for prime p), A145199.

[ p: p in PrimesUpTo(500) | not IsSquarefree(p-1) ]; // Vincenzo Librandi, Mar 12 2015

Select[Prime[Range[400]], MoebiusMu[ #-1]==0&]

forprime(p=2,500,if(!issquarefree(p-1),print(p))) \\ Michael B. Porter, Mar 16 2015

a(n) = A145199(n) + 1. - Amiram Eldar, Feb 10 2021

A049149 Numbers k such that the Euler totient function phi(k) is squarefree.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 11, 14, 18, 22, 23, 31, 43, 46, 47, 49, 59, 62, 67, 71, 79, 83, 86, 94, 98, 103, 107, 118, 121, 131, 134, 139, 142, 158, 166, 167, 179, 191, 206, 211, 214, 223, 227, 239, 242, 262, 263, 278, 283, 311, 331, 334, 347, 358, 359, 367, 382, 383

Offset: 1

Labos Elemer

nonn

Consists of 1, 2, 4, p, p^2, 2p, and 2p^2, where p are the odd primes from A039787. - Ivan Neretin, Aug 24 2016

			a(17) = 49 is here because phi(49) = 42 = 2*3*7 is squarefree. Primes p, such that p-1 is squarefree are included.

Ivan Neretin, Table of n, a(n) for n = 1..10000
William D. Banks and Francesco Pappalardi, Values of the Euler function free of kth powers, Journal of Number Theory, Vol. 120, No. 2 (2006), pp. 326-348.
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. A000010, A000720, A005117, A005596, A039787, A013929.

Select[Range[100], MoebiusMu[EulerPhi[#]] != 0 &]

isok(n) = issquarefree(eulerphi(n)); \\ Michel Marcus, Aug 24 2016

The number of terms not exceeding k is (3*a/2) * pi(k) + O(k/(log(k)^c)), where pi(k) = A000720(k), c is any constant > 0, and a = 0.373955... is Artin's constant (A005596) (Pappalardi et al., 2003; Banks and Pappalardi, 2006). - Amiram Eldar, Jul 28 2020

Corrected by T. D. Noe, Oct 25 2006

A241194 Numerator of phi(p-1)/(p-1), where phi is Euler's totient function and p = prime(n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 4, 1, 2, 2, 11, 6, 14, 4, 10, 12, 1, 4, 20, 5, 1, 2, 16, 26, 1, 3, 2, 24, 8, 22, 18, 4, 4, 1, 41, 21, 44, 4, 36, 1, 3, 10, 8, 12, 56, 6, 14, 48, 4, 2, 1, 65, 33, 4, 22, 12, 46, 36, 16, 12, 4, 39, 8, 2, 86, 28, 5, 89, 20, 10, 2, 95

Offset: 1

T. D. Noe, Apr 17 2014

The denominators are in A241195. The new minima of phi(p-1)/(p-1) occur at primes listed in A241196. The numerator and denominator of those terms are in A241197 and A241198.

For primes p>2, the fraction phi(p - 1)/(p - 1) has the maximum value = 1/2 if and only if p is in A019434. - Geoffrey Critzer, Dec 30 2014

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 117.

T. D. Noe, Table of n, a(n) for n = 1..10000
P. Erdős, On the density of some sequences of numbers, III., J. London Math. Soc. 13 (1938), pp. 119-127.
Imre Kátai, On distribution of arithmetical functions on the set prime plus one, Compositio Math. 19 (1968), pp. 278-289.
I. J. Schoenberg, Über die asymptotische Verteilung reeller Zahlen mod 1, Mathematische Zeitschrift 28:1 (1928), pp. 171-199.

Cf. A000010, A005596, A008330 (phi(prime(n)-1)), A065485, A241195, A241196, A241197, A241198.

Cf. A076512, A006093.

[Numerator(EulerPhi(NthPrime(n)-1)/(NthPrime(n)-1)): n in [1..80]]; // Vincenzo Librandi, Apr 06 2015

seq(numer(numtheory:-phi(ithprime(i)-1)/(ithprime(i)-1)), i=1..100); # Robert Israel, Jan 11 2015

Numerator[Table[EulerPhi[p - 1]/(p - 1), {p, Prime[Range[100]]}]]

lista(nn) = forprime(p=2, nn, print1(numerator(eulerphi(p-1)/(p-1)), ", ")); \\ Michel Marcus, Jan 03 2015

From Amiram Eldar, Jul 31 2020: (Start)
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{n=1..m} a(n)/A241195(n) = 0.373955... (Artin's constant, A005596).
Asymptotic mean of inverse ratio: lim_{m->oo} (1/m) * Sum_{n=1..m} A241195(n)/a(n) = 2.826419... (Murata's constant, A065485). (End)
a(n) = A076512(A006093(n)). - Ridouane Oudra, Mar 24 2025

A377177 Primes p such that -7/2 is a primitive root modulo p.

Original entry on oeis.org

11, 17, 29, 31, 37, 41, 43, 47, 73, 89, 103, 107, 109, 149, 167, 179, 197, 257, 277, 311, 313, 317, 347, 353, 367, 373, 383, 389, 409, 433, 479, 491, 499, 503, 521, 541, 557, 571, 577, 593, 601, 607, 647, 653, 659, 683, 701, 719, 727, 761, 769, 821, 839, 857, 883, 887, 907, 929, 937, 947, 983

Offset: 1

Jianing Song, Oct 18 2024

If p is a term in this sequence, then -7/2 is not a square modulo p (i.e., p is in A191061).

Conjecture: this sequence has relative density equal to Artin's constant (A005596) with respect to the set of primes.

Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Artin's conjecture on primitive roots.
Index entries for primes by primitive root

Primes p such that +a/2 is a primitive root modulo p: A320384 (a=3), A377174 (a=5), A377176 (a=7), A377178 (a=9).
Primes p such that -a/2 is a primitive root modulo p: A377172 (a=3), A377175 (a=5), this sequence (a=7), A377179 (a=9).
Cf. A191061, A005596.

select(p -> isprime(p) and numtheory:-order(-7/2 mod p, p) = p-1, [seq(i,i=3..1000,2)]); # Robert Israel, Nov 10 2024

Cases[Prime[Range[2,170]],?(MemberQ[PrimitiveRootList[#],ResourceFunction["FractionMod"][-7/2,#]]&)] (* _Shenghui Yang, Oct 23 2024 *)

forprime(p=8,10^3,if(znorder(Mod(-7/2,p))==p-1,print1(p,", "))); \\ Joerg Arndt, Oct 19 2024

cp's OEIS Frontend

A049098 Primes p such that p+1 is divisible by a square.

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A065415 Decimal expansion of Product_{p prime} (1-1/(p^4-p^3)).

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A065416 Decimal expansion of Product_{p prime} (1-1/(p^5-p^4)).

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A065488 Decimal expansion of Product_{p prime} (1 + 1/(p^2-p-1)).

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A019338 Primes with primitive root 8.

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A048296 Continued fraction for Artin's constant.

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A049092 Primes p such that p-1 is not squarefree.

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