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A293756 a(n) = smallest number k with n prime factors such that d + k/d is prime for every d | k.

%I A293756 #73 Feb 16 2025 08:33:51
%S A293756 1,2,6,30,210,186162
%N A293756 a(n) = smallest number k with n prime factors such that d + k/d is prime for every d | k.
%C A293756 For n > 0, a(n) is even and squarefree.
%C A293756 For n > 0, a(n) gives 2^(n-1) distinct primes.
%C A293756 If the k-tuple conjecture is true, then this sequence is infinite. - Carl Pomerance, Nov 12 2017
%C A293756 a(n) is the least integer k with n prime divisors such that A282849(k) = A000005(k). - _Michel Marcus_, Nov 13 2017
%C A293756 a(n) is the smallest k with n prime factors such that A282849(k) = 2^n. - _Thomas Ordowski_, Nov 13 2017
%C A293756 a(6), if it exists, has a prime divisor greater than 10^3. - _Arkadiusz Wesolowski_, Nov 14 2017
%H A293756 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/k-TupleConjecture.html">k-Tuple Conjecture</a>
%F A293756 a(n) = 2*A295124(n-1) for n > 0. - _Thomas Ordowski_, Nov 15 2017
%e A293756 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.
%e A293756 From _Michael De Vlieger_, Nov 14 2017: (Start)
%e A293756 First differences of prime indices of a(n):
%e A293756 n       a(n)   A287352(a(n))
%e A293756 -----------------------------
%e A293756 1         2    1
%e A293756 2         6    1, 1
%e A293756 3        30    1, 1, 1
%e A293756 4       210    1, 1, 1, 1
%e A293756 5    186162    1, 1, 6, 1, 11
%e A293756 (End)
%p A293756 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;
%p A293756 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
%o A293756 (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;
%o A293756 a(n) = {my(k=1); while( !isok(k, n), k++); k;} \\ _Michel Marcus_, Nov 11 2017
%Y A293756 Subsequence of A080715 (d + k/d is prime for every d|k).
%Y A293756 Cf. A103787, A282849, A294925, A295124, A295169.
%K A293756 nonn,more
%O A293756 0,2
%A A293756 _Thomas Ordowski_, Nov 11 2017
%E A293756 a(5) from _Michel Marcus_, Nov 11 2017