A215504 -id:A215504 - cp's OEIS frontend

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.

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]

A343479 Prime numbers p == 2 (mod 3) such that p-1 has exactly one odd prime divisor and p+1 has exactly one prime divisor > 3 (counting prime divisors with multiplicity in both).

Original entry on oeis.org

29, 41, 59, 83, 89, 113, 137, 167, 173, 179, 227, 233, 263, 269, 317, 347, 353, 359, 467, 479, 503, 557, 563, 593, 641, 653, 719, 773, 809, 887, 977, 983, 1097, 1187, 1193, 1283, 1307, 1367, 1433, 1439, 1487, 1493, 1523, 1619, 1697, 1823, 1907, 1997, 2063, 2153

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 odd prime divisor (7) and 29+1 = 30 = 2*3*5 has only one prime divisor (5) 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.

Subsequence of A343478.

Cf. A003627, A038550, A215504, A343480.

q[n_] := Mod[n, 3] == 2 && PrimeQ[n] && PrimeQ[(n + 1)/2^IntegerExponent[n + 1, 2]/3^IntegerExponent[n + 1, 3]] && PrimeQ[(n - 1)/2^IntegerExponent[n - 1, 2]]; Select[Range[2000], q]

A284037 Primes p such that p-1 and p+1 have two distinct prime factors.

Original entry on oeis.org

11, 13, 19, 23, 37, 47, 53, 73, 97, 107, 163, 193, 383, 487, 577, 863, 1153, 2593, 2917, 4373, 8747, 995327, 1492993, 1990657, 5308417, 28311553, 86093443, 6879707137, 1761205026817, 2348273369087, 5566277615617, 7421703487487, 21422803359743, 79164837199873

Offset: 1

Giuseppe Coppoletta, Mar 28 2017

nonn

Either p-1 or p+1 must be of the form 2^i * 3^j, since among three consecutive numbers exactly one is a multiple of 3. - Giovanni Resta, Mar 29 2017

Subsequence of A219528. See the previous comment. - Jason Yuen, Mar 08 2025

			7 is not a term because n + 1 = 8 has only one prime factor.
23 is a term because it is prime and n - 1 = 22 has two distinct prime factors (2, 11) and n + 1 = 24 has two distinct prime factors (2, 3).
43 is not a term because n - 1 = 42 has three distinct prime factors (2, 3, 7).

Robert Israel, Table of n, a(n) for n = 1..51 (all terms < 10^75)

Cf. A000668, A001221, A005105, A005109, A033845, A058383, A067386, A092506, A215504, A219528, A275598.

N:= 10^20: # To get all terms <= N
Res:= {}:
for i from 1 to ilog2(N) do
  for j from 1 to floor(log[3](N/2^i)) do
    q:= 2^i*3^j;
    if isprime(q-1) and nops(numtheory:-factorset((q-2)/2^padic:-ordp(q-2,2)))=1 then Res:= Res union {q-1} fi;
    if isprime(q+1) and nops(numtheory:-factorset((q+2)/2^padic:-ordp(q+2,2)))=1 then Res:= Res union {q+1} fi
od od:
sort(convert(Res,list)); # Robert Israel, Apr 16 2017

mx = 10^30; ok[t_] := PrimeQ[t] && PrimeNu[t-1]==2==PrimeNu[t+1]; Sort@ Reap[Do[ w = 2^i 3^j; Sow /@ Select[ w+ {1,-1}, ok], {i, Log2@ mx}, {j, 1, Log[3, mx/2^i]}]][[2, 1]] (* terms up to mx, Giovanni Resta, Mar 29 2017 *)

isok(n) = isprime(n) && (omega(n-1)==2) && (omega(n+1)==2); \\ Michel Marcus, Apr 17 2017

omega=sloane.A001221; [n for n in prime_range(10^6) if 2==omega(n-1)==omega(n+1)]

sorted([2^i*3^j+k for i in (1..40) for j in (1..20) for k in (-1,1) if is_prime(2^i*3^j+k) and sloane.A001221(2^i*3^j+2*k)==2])

A001221(a(n)) = 1 and A001221(a(n) - 1) = A001221(a(n) + 1) = 2.

a(33)-a(34) from Giovanni Resta, Mar 29 2017

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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.

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A343479 Prime numbers p == 2 (mod 3) such that p-1 has exactly one odd prime divisor and p+1 has exactly one prime divisor > 3 (counting prime divisors with multiplicity in both).

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A284037 Primes p such that p-1 and p+1 have two distinct prime factors.

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