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URL: https://oeis.org/A284037

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A284037
Primes p such that p-1 and p+1 have two distinct prime factors.
2
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
Subsequence of A219528. See the previous comment. - Jason Yuen, Mar 08 2025
LINKS
Robert Israel, Table of n, a(n) for n = 1..51 (all terms < 10^75)
FORMULA
A001221(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
(SageMath) omega=sloane.A001221; [n for n in prime_range(10^6) if 2==omega(n-1)==omega(n+1)]
(SageMath) 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
KEYWORD
nonn
AUTHOR
Giuseppe Coppoletta, Mar 28 2017
EXTENSIONS
a(33)-a(34) from Giovanni Resta, Mar 29 2017
STATUS
approved