A343479 -id:A343479 - cp's OEIS frontend

A343480 Decimal expansion of 9 * Product_{p prime >= 5} (1-3/p)/(1-1/p)^3.

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

Offset: 1

Amiram Eldar, Apr 16 2021

This constant appears in the conjectured asymptotic formulas for A343478 and A343479.

			5.71649719143844086486026932145270175607859118599135...

Carlos Esparza and Lukas Gehring, Estimating the density of a set of primes with applications to group theory, arXiv:1810.08679 [math.NT], 2018.

Cf. A271886, A343478, A343479.

9 * prodeulerrat((1-3/p)/(1-1/p)^3, 1, 5)

Equals 2*A271886. - Hugo Pfoertner, Feb 11 2025

A343478 Prime numbers p == 2 (mod 3) such that p-1 has exactly one distinct odd prime divisor and p+1 has exactly one distinct prime divisor > 3.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Apr 16 2021

nonn

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

			29 is a term since it is prime, 29 = 3*9 + 2, 29-1 = 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, 101-1 = 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.

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.

A343479 is a subsequence.

Cf. A003627, A215504, A336101, A343480.

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]

cp's OEIS Frontend

A343480 Decimal expansion of 9 * Product_{p prime >= 5} (1-3/p)/(1-1/p)^3.

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