A293756 a(n) = smallest number k with n prime factors such that d + k/d is prime for every d | k.
1, 2, 6, 30, 210, 186162
Offset: 0
Examples
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)
Links
- Eric Weisstein's World of Mathematics, k-Tuple Conjecture
Crossrefs
Programs
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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
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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
Formula
a(n) = 2*A295124(n-1) for n > 0. - Thomas Ordowski, Nov 15 2017
Extensions
a(5) from Michel Marcus, Nov 11 2017
Comments