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

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

1,1

Comments

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

Links

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

Formula

A001222(a(n)) = 1 and A001221(a(n)) - 1) = A001221(a(n)) + 1) = 2.

Example

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).

Maple

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

Mathematica

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 *)

Prog

(Sage) omega=sloane.A001221; [n for n in prime_range(10^6) if 2==omega(n-1)==omega(n+1)]
(Sage) 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])
(PARI) isok(n) = isprime(n) && (omega(n-1)==2) && (omega(n+1)==2); \\ Michel Marcus, Apr 17 2017

Crossrefs

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

Sequence in context: A215504 A342406 A109650 * A338570 A321891 A153421

Adjacent sequences: A284034 A284035 A284036 * A284038 A284039 A284040

Keyword

nonn

Author

Giuseppe Coppoletta, Mar 28 2017

Extensions

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

Status

approved