OFFSET
0,2
COMMENTS
For n > 0, a(n) is even and squarefree.
For n > 0, a(n) gives 2^(n-1) distinct primes.
If the k-tuple conjecture is true, then this sequence is infinite. - Carl Pomerance, Nov 12 2017
a(n) is the least integer k with n prime divisors such that A282849(k) = A000005(k). - Michel Marcus, Nov 13 2017
a(n) is the smallest k with n prime factors such that A282849(k) = 2^n. - Thomas Ordowski, Nov 13 2017
a(6), if it exists, has a prime divisor greater than 10^3. - Arkadiusz Wesolowski, Nov 14 2017
LINKS
Eric Weisstein's World of Mathematics, k-Tuple Conjecture
FORMULA
a(n) = 2*A295124(n-1) for n > 0. - Thomas Ordowski, Nov 15 2017
EXAMPLE
a(2) = 2*3 = 6 because k = 6 is the smallest number with 2 prime factors such that for d = {1, 2, 3, 6} we have 1 + 6/1 = 6 + 6/6 = 7 is prime and 2 + 6/2 = 3 + 6/3 = 5 is prime.
From Michael De Vlieger, Nov 14 2017: (Start)
First differences of prime indices of a(n):
n a(n) A287352(a(n))
-----------------------------
1 2 1
2 6 1, 1
3 30 1, 1, 1
4 210 1, 1, 1, 1
5 186162 1, 1, 6, 1, 11
(End)
MAPLE
with(numtheory): P:=proc(q) local a, b, j, k, n, ok; print(1); for n from 1 to q do for k from 2 to q do a:=ifactors(k)[2]; a:=add(a[j][2], j=1..nops(a)); if a=n then b:=divisors(k); ok:=1;
for j from 1 to nops(b) do if not isprime(b[j]+k/b[j]) then ok:=0; break; fi; od; if ok=1 then print(k); break; fi; fi; od; od; end: P(10^8); # Paolo P. Lava, Nov 16 2017
PROG
(PARI) isok(k, n)=if (!issquarefree(k), return (0)); if (omega(k) != n, return (0)); fordiv(k, d, if (!isprime(d+k/d), return(0))); 1;
a(n) = {my(k=1); while( !isok(k, n), k++); k; } \\ Michel Marcus, Nov 11 2017
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Thomas Ordowski, Nov 11 2017
EXTENSIONS
a(5) from Michel Marcus, Nov 11 2017
STATUS
approved