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A284037 Primes p such that p-1 and p+1 have two distinct prime factors. 1
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 (list; graph; refs; listen; history; text; internal format)
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

with(numtheory): P:=proc(q);

if nops(ifactors(ithprime(q)-1)[2])=2 and nops(ifactors(ithprime(q)+1)[2])=2

then ithprime(q); fi; end: seq(P(i), i=1..10^5); # Paolo P. Lava, Mar 29 2017

# Alternative:

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: A068801 A215504 A109650 * A321891 A153421 A050265

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

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Last modified January 27 04:57 EST 2020. Contains 331291 sequences. (Running on oeis4.)