A106849 -id:A106849 - cp's OEIS frontend

A221981 Primes q = 4*p+1, where p == 2 (mod 5) is also prime.

29, 149, 269, 389, 509, 1109, 1229, 1949, 2309, 2909, 3989, 4349, 5189, 5309, 6269, 6389, 7109, 7949, 8069, 9749, 10589, 10709, 11069, 11549, 12149, 12269, 13229, 13829, 14549, 15629, 16229, 17189, 17789, 18269, 19949, 20789, 22109, 22229, 24029, 24989, 25349, 25469, 25589, 26189, 26309, 28109, 28229, 28949, 29669, 30029, 30869, 31469, 32069, 33149, 34589, 34949, 36269, 36629, 36749, 37589

Offset: 1

Jonathan Sondow, Feb 02 2013

nonn

Moree (2012) says that Chebyshev observed that if q = 4p + 1 is prime, with prime p == 2 (mod 5), then 10 is a primitive root modulo q.
If the sequence is infinite, then Artin's conjecture ("every nonsquare integer n != -1 is a primitive root of infinitely many primes q") is true for n = 10.
The corresponding primes p are A221982.
The sequence is infinite under Dickson's conjecture, thus Dickson's conjecture implies Artin's conjecture for n = 10. - Charles R Greathouse IV, Apr 18 2013
Two conjectures: (a) These primes have primitive root 40; (b) if a(n)*8 + 1 is prime then it has primitive root 10. - Davide Rotondo, Dec 31 2024

			29 is a member because 29 = 4*7 + 1 and 7 == 2 (mod 5) are prime.

P. L. Chebyshev, Theory of congruences, Elements of number theory, Chelsea, 1972, p. 306.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section F9, pp. 377-380.

Amiram Eldar, Table of n, a(n) for n = 1..10000
P. L. Chebyshev, Theorie der Congruenzen, Mayer & Mueller, 1889, pp. 306-313.
Pieter Moree, Artin's primitive root conjecture - a survey, arXiv:math/0412262 [math.NT], 2004, revised 2012, p. 1.
Index entries for sequences related to Artin's conjecture
Index entries for primes by primitive root

Cf. A001913, A005596, A006883, A045380, A106849, A221982, A222008.

A221981:=n->`if`(isprime(4*n+1) and isprime(n) and n mod 5 = 2, 4*n+1, NULL): seq(A221981(n), n=1..10^4); # Wesley Ivan Hurt, Dec 11 2015

Select[ Prime[ Range[4000]], Mod[(# - 1)/4, 5] == 2 && PrimeQ[(# - 1)/4] &]

is(n)=n%20==9 && isprime(n) && isprime(n\4) \\ Charles R Greathouse IV, Apr 18 2013

a(n) = 4*A221982(n) + 1.

a(n) >> n log^2 n. - Charles R Greathouse IV, Dec 30 2024

A221982 Primes p == 2 (mod 5) for which 4*p+1 is also prime.

Original entry on oeis.org

7, 37, 67, 97, 127, 277, 307, 487, 577, 727, 997, 1087, 1297, 1327, 1567, 1597, 1777, 1987, 2017, 2437, 2647, 2677, 2767, 2887, 3037, 3067, 3307, 3457, 3637, 3907, 4057, 4297, 4447, 4567, 4987, 5197, 5527, 5557, 6007, 6247, 6337, 6367, 6397, 6547, 6577, 7027, 7057, 7237, 7417, 7507, 7717, 7867

Offset: 1

Jonathan Sondow, Feb 02 2013

nonn

The corresponding primes 4*p+1 are Chebyshev's subsequence A221981 of the primes with primitive root 10.

			7 is a member because 7 == 2 (mod 5) and 29 = 4*7 + 1 are both prime.

P. L. Chebyshev, Theory of congruences. Elements of number theory, Chelsea, 1972, p. 306.
R. K. Guy, Unsolved Problems in Number Theory, F9.

Paolo P. Lava, Table of n, a(n) for n = 1..10000
P. Moree, Artin's primitive root conjecture - a survey, arXiv 2004, revised 2012, p. 1.
Index entries for primes by primitive root

Cf. A001913, A006883, A045380, A106849, A221981, A222008.

A221982:=proc(q)
local n;
for n from 1 to q do
if isprime(n) and isprime(4*n+1) and (n mod 5)=2 then print(n) fi; od; end:
A221982 (10000); # Paolo P. Lava, Feb 12 2013

Select[ Prime[ Range[1000]], Mod[#, 5] == 2 && PrimeQ[4 # + 1] &]

a(n) = (A221981(n) - 1)/4.

A106848 a(n) = the number of times the last digit of n must be appended to n to form a number m such that n divides m, or 0 if no such m exists.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 6, 6, 3, 0, 16, 9, 18, 1, 6, 2, 22, 0, 0, 6, 27, 6, 28, 1, 15, 0, 2, 16, 6, 0, 3, 18, 6, 1, 5, 6, 21, 2, 9, 22, 46, 0, 42, 1, 48, 0, 13, 27, 2, 0, 18, 28, 58, 1, 60, 15, 6, 0, 6, 2, 33, 16, 22, 1, 35, 0, 8, 3, 0, 0, 2, 6, 13, 1, 81, 5, 41, 6, 16, 21, 84, 2, 44, 1

Offset: 1

Chuck Seggelin (seqfan(AT)plastereddragon.com), May 08 2005

Shares many terms in common with A064696, which involved inserting zeros between digits. Numbers which do not appear to be able to form a multiple (a(n)=0) were tested out to 10000 digits added. Note those values of n for which a(n)=0 (12, 16, 24, 25, 32, 36, 48, ...) appear to be given by A064695.

			a(13) = 6 because the last digit of 13 must be appended to it six times before a new number which divides 13 is formed. (I.e., 133 mod 13 = 3, 1333 mod 13 = 7, 13333 mod 13 = 8, 133333 mod 13 = 5, 1333333 mod 13 = 6, 13333333 mod 13 = 0.) a(12)=0 because no matter how many 2's are appended to 12, the resulting number is not divisible by 12.

Cf. A106849.

cp's OEIS Frontend

A221981 Primes q = 4*p+1, where p == 2 (mod 5) is also prime.

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A221982 Primes p == 2 (mod 5) for which 4*p+1 is also prime.

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A106848 a(n) = the number of times the last digit of n must be appended to n to form a number m such that n divides m, or 0 if no such m exists.

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