

A343478


Prime numbers p == 2 (mod 3) such that p1 has exactly one distinct odd prime divisor and p+1 has exactly one distinct prime divisor > 3.


3



29, 41, 59, 83, 89, 101, 113, 137, 149, 167, 173, 179, 197, 227, 233, 251, 263, 269, 293, 317, 347, 353, 359, 401, 449, 467, 479, 503, 557, 563, 587, 593, 641, 653, 677, 719, 773, 809, 887, 977, 983, 1097, 1187, 1193, 1283, 1307, 1367, 1373, 1433, 1439, 1487, 1493
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OFFSET

1,1


COMMENTS

Esparza and Gehring (2018) proved that assuming a generalized HardyLittlewood conjecture the number of terms not exceeding x is asymptotically (c/2) * x/log(x)^3, where c = A343480 = 5.716497...


LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000
Carlos Esparza and Lukas Gehring, Estimating the density of a set of primes with applications to group theory, arXiv:1810.08679 [math.NT], 2018.


EXAMPLE

29 is a term since it is prime, 29 = 3*9 + 2, 291 = 28 = 2^2 * 7 has only one distinct odd prime divisor (7) and 29+1 = 30 = 2*3*5 has only one distinct prime divisor (5) larger than 3.
101 is a term since it is prime, 101 = 3*33 + 2, 1011 = 100 = 2^2 * 5^2 has only one distinct odd prime divisor (5) and 101+1 = 102 = 2^2*3*17 has only one distinct prime divisor (17) larger than 3.


MATHEMATICA

q[n_] := Mod[n, 3] == 2 && PrimeQ[n] && PrimeNu[(n + 1)/2^IntegerExponent[n + 1, 2]/3^IntegerExponent[n + 1, 3]] == 1 && PrimeNu[(n  1)/2^IntegerExponent[n  1, 2]] == 1; Select[Range[1500], q]


CROSSREFS

A343479 is a subsequence.
Cf. A003627, A215504, A336101, A343480.
Sequence in context: A161616 A069454 A035789 * A343479 A080899 A216815
Adjacent sequences: A343475 A343476 A343477 * A343479 A343480 A343481


KEYWORD

nonn


AUTHOR

Amiram Eldar, Apr 16 2021


STATUS

approved



