A005596 -id:A005596 - cp's OEIS frontend

A049233 Primes p such that p + 2 is squarefree.

3, 5, 11, 13, 17, 19, 29, 31, 37, 41, 53, 59, 67, 71, 83, 89, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 157, 163, 179, 181, 191, 193, 197, 199, 211, 227, 229, 233, 239, 251, 257, 263, 269, 271, 281, 283, 293, 307, 311, 317, 337, 347, 353, 379, 383, 389

Offset: 1

Labos Elemer

nonn

A001359 (lesser of twin primes) is a subsequence. - Michel Marcus, Aug 10 2018

This sequence is infinite and its relative density in the sequence of primes is equal to 2 * Product_{p prime} (1-1/(p*(p-1))) = 2 * A005596 = 0.747911... (Mirsky, 1949). - Amiram Eldar, Dec 29 2020

Robert Israel, Table of n, a(n) for n = 1..10000
Denis Xavier Charles, Sieve Methods, Master's Thesis, 2000, p. 93.
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. A001359, A005117, A005596.

A049233:=n->`if`(isprime(n) and numtheory[issqrfree](n+2), n, NULL): seq(A049233(n), n=1..600); # Wesley Ivan Hurt, Nov 22 2015

Select[Prime[Range[100]], SquareFreeQ[#+2]&] (* Jean-François Alcover, Nov 22 2015 *)

isok(p) = isprime(p) && issquarefree(p+2); \\ Michel Marcus, Dec 31 2013

Definition simplified by Michel Marcus, Dec 31 2013

A077064 Squarefree numbers of form prime - 1.

Original entry on oeis.org

1, 2, 6, 10, 22, 30, 42, 46, 58, 66, 70, 78, 82, 102, 106, 130, 138, 166, 178, 190, 210, 222, 226, 238, 262, 282, 310, 330, 346, 358, 366, 382, 418, 430, 438, 442, 462, 466, 478, 498, 502, 546, 562, 570, 586, 598, 606, 618, 642, 646, 658, 682, 690, 718, 742

Offset: 1

Reinhard Zumkeller, Oct 23 2002

nonn

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... (Victorovich, 2013). - Amiram Eldar, Dec 29 2020

			A005117(44) = 70 = 2*5*7 is a term as 70 = A000040(20)-1 = 71-1.

Harvey P. Dale, Table of n, a(n) for n = 1..10000
Radoslav Tsvetkov, On the distribution of k-free numbers and r-tuples of k-free numbers. A survey, Notes on Number Theory and Discrete Mathematics, Vol. 25, No. 3 (2019), pp. 207-222. See section 3.4, p. 210.
G. D. Victorovich, On additive property of arithmetic functions (in Russian), Thesis, Moscow State University, 2013.

Cf. A005596, A006093, A005117, A000040, A077063, A077067.

Select[Prime[Range[200]]-1,SquareFreeQ] (* Harvey P. Dale, Feb 09 2015 *)

isok(n) = issquarefree(n) && isprime(n+1); \\ Michel Marcus, Mar 22 2016

lista(nn) = forprime(p=2, nn, if (issquarefree(p-1), print1(p-1, ", "))); \\ Michel Marcus, Mar 22 2016

Wrong formula removed by Amiram Eldar, Dec 29 2020

A377172 Primes p such that -3/2 is a primitive root modulo p.

Original entry on oeis.org

17, 23, 37, 41, 43, 47, 67, 89, 109, 113, 137, 139, 157, 163, 167, 191, 229, 233, 239, 257, 263, 277, 283, 311, 349, 353, 359, 379, 383, 397, 421, 449, 479, 503, 521, 523, 541, 547, 569, 571, 593, 599, 613, 619, 641, 647, 661, 719, 733, 739, 743, 757, 761, 787, 809, 811, 839, 853, 857, 859, 863, 877, 887, 911, 929, 953, 977, 983

Offset: 1

Jianing Song, Oct 18 2024

If p is a term in this sequence, then -3/2 is not a square modulo p (i.e., p is in A191059).

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

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: this sequence (a=3), A377175 (a=5), A377177 (a=7), A377179 (a=9).
Cf. A191059, A005596.

forprime(p=5, 10^3, if(znorder(Mod(-3/2, p))==p-1, print1(p, ", ")));

A377174 Primes p such that 5/2 is a primitive root modulo p.

Original entry on oeis.org

11, 17, 23, 47, 59, 73, 101, 103, 109, 113, 137, 139, 149, 167, 179, 211, 223, 229, 233, 257, 263, 269, 313, 337, 349, 353, 367, 379, 383, 389, 419, 421, 433, 461, 487, 499, 503, 509, 593, 607, 617, 647, 659, 661, 673, 727, 743, 811, 821, 823, 829, 857, 859, 863, 887, 941, 953, 967, 971, 977, 983

Offset: 1

Jianing Song, Oct 18 2024

If p is a term in this sequence, then 5/2 is not a square modulo p (i.e., p is in A038880).

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

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), this sequence (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), A377177 (a=7), A377179 (a=9).
Cf. A038880, A005596.

forprime(p=7, 10^3, if(znorder(Mod(5/2, p))==p-1, print1(p, ", ")));

A377175 Primes p such that -5/2 is a primitive root modulo p.

Original entry on oeis.org

3, 17, 31, 43, 67, 71, 73, 79, 83, 101, 107, 109, 113, 137, 149, 163, 191, 199, 227, 229, 233, 239, 257, 269, 271, 283, 307, 311, 313, 337, 347, 349, 353, 359, 389, 421, 431, 433, 439, 443, 461, 467, 479, 509, 547, 563, 587, 593, 599, 617, 631, 661, 673, 683, 719, 821, 827, 829, 839, 857, 907, 911, 919, 941, 947, 953, 977

Offset: 1

Jianing Song, Oct 18 2024

If p is a term in this sequence, then -5/2 is not a square modulo p (i.e., p is in A296925).

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

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), this sequence (a=5), A377177 (a=7), A377179 (a=9).
Cf. A296925, A005596.

print1(3, ", "); forprime(p=7, 10^3, if(znorder(Mod(-5/2, p))==p-1, print1(p, ", ")));

A377176 Primes p such that 7/2 is a primitive root modulo p.

Original entry on oeis.org

3, 17, 19, 23, 29, 37, 41, 59, 73, 79, 83, 89, 109, 127, 139, 149, 191, 197, 227, 239, 251, 257, 263, 277, 283, 307, 313, 317, 353, 359, 373, 389, 409, 419, 431, 433, 467, 487, 521, 523, 541, 557, 563, 577, 587, 593, 599, 601, 619, 643, 653, 691, 701, 761, 769, 821, 857, 863, 919, 929, 937, 967, 991

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 A038886).

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

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), this sequence (a=7), A377178 (a=9).
Primes p such that -a/2 is a primitive root modulo p: A377172 (a=3), A377175 (a=5), A377177 (a=7), A377179 (a=9).
Cf. A038886, A005596.

print1(3, ", "); forprime(p=11, 10^3, if(znorder(Mod(-5/2, p))==p-1, print1(p, ", ")));

A377178 Primes p such that 9/2 is a primitive root modulo p.

Original entry on oeis.org

5, 13, 19, 29, 43, 53, 59, 61, 83, 101, 107, 109, 149, 157, 173, 179, 197, 227, 229, 251, 269, 277, 283, 293, 317, 331, 347, 373, 389, 419, 443, 461, 467, 491, 509, 523, 547, 557, 563, 587, 613, 619, 653, 661, 677, 683, 691, 701, 709, 733, 739, 757, 773, 787, 797, 821, 829, 853, 883, 907, 947, 971

Offset: 1

Jianing Song, Oct 18 2024

If p is a term in this sequence, then 9/2 is not a square modulo p (i.e., p is in A003629).

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

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), this sequence (a=9).
Primes p such that -a/2 is a primitive root modulo p: A377172 (a=3), A377175 (a=5), A377177 (a=7), A377179 (a=9).
Cf. A003629, A005596.

forprime(p=5, 10^3, if(znorder(Mod(9/2, p))==p-1, print1(p, ", ")));

A377179 Primes p such that -9/2 is a primitive root modulo p.

Original entry on oeis.org

5, 13, 23, 29, 31, 47, 53, 61, 71, 79, 101, 109, 149, 151, 157, 167, 173, 191, 197, 199, 223, 229, 239, 263, 269, 277, 293, 311, 317, 359, 367, 373, 383, 389, 461, 463, 479, 487, 503, 509, 557, 599, 613, 647, 653, 661, 677, 701, 709, 719, 733, 743, 757, 773, 797, 821, 823, 829, 839, 853, 863, 887, 911, 967, 983, 991

Offset: 1

Jianing Song, Oct 18 2024

If p is a term in this sequence, then -9/2 is not a square modulo p (i.e., p is in A003628).

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

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), A377177 (a=7), this sequence (a=9).
Cf. A003628, A005596.

forprime(p=5, 10^3, if(znorder(Mod(-9/2, p))==p-1, print1(p, ", ")));

A007348 Primes for which -10 is a primitive root.

Original entry on oeis.org

3, 17, 29, 31, 43, 61, 67, 71, 83, 97, 107, 109, 113, 149, 151, 163, 181, 191, 193, 199, 227, 229, 233, 257, 269, 283, 307, 311, 313, 337, 347, 359, 389, 431, 433, 439, 443, 461, 467, 479, 509, 523, 541, 563, 577, 587, 593, 599, 631, 683, 701, 709, 719, 787, 821, 827, 839

Offset: 1

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

nonn

Union of long period primes (A006883) of the form 4k+1 and half period primes (A097443) of the form 4k+3. - Davide Rotondo, Aug 25 2021

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), Table 24.8, p. 864.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. J. Mathar, Table of n, a(n) for n = 1..10000
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. A038880.

Cf. A006883, A097443.

pr=-10; Select[Prime[Range[200 ] ], MultiplicativeOrder[pr, # ] == #-1 & ]

is(n)=gcd(n,10)==1 && znorder(Mod(-10,n))==n-1 \\ Charles R Greathouse IV, Nov 25 2014

More terms from N. J. A. Sloane, Apr 24 2005
Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar
A&S reference and Mathematica program corrected by T. D. Noe, Nov 04 2009

A049231 Primes p such that p - 2 is squarefree.

Original entry on oeis.org

3, 5, 7, 13, 17, 19, 23, 31, 37, 41, 43, 53, 59, 61, 67, 71, 73, 79, 89, 97, 103, 107, 109, 113, 131, 139, 151, 157, 163, 167, 179, 181, 193, 197, 199, 211, 223, 229, 233, 239, 241, 251, 257, 269, 271, 283, 293, 307, 311, 313, 331, 337, 347, 349, 359, 367, 373

Offset: 1

Labos Elemer

nonn

This sequence is infinite and its relative density in the sequence of the primes is equal to 2 * Product_{p prime} (1-1/(p*(p-1))) = 2 * A005596 = 0.747911... (Mirsky, 1949). - Amiram Eldar, Feb 27 2021

Seiichi Manyama, 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. A005117, A005596, A039787, A049233.

Select[Prime[Range[100]],SquareFreeQ[#-2]&] (* Harvey P. Dale, Mar 03 2018 *)

isok(p) = isprime(p) && issquarefree(p-2); \\ Michel Marcus, Dec 31 2013

Primes p such that abs(mu(p-2)) = 1.

Definition corrected by Michel Marcus, Dec 31 2013

cp's OEIS Frontend

A049233 Primes p such that p + 2 is squarefree.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A077064 Squarefree numbers of form prime - 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A377172 Primes p such that -3/2 is a primitive root modulo p.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A377174 Primes p such that 5/2 is a primitive root modulo p.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A377175 Primes p such that -5/2 is a primitive root modulo p.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A377176 Primes p such that 7/2 is a primitive root modulo p.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A377178 Primes p such that 9/2 is a primitive root modulo p.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A377179 Primes p such that -9/2 is a primitive root modulo p.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI