A356801 a(n) is the least semiprime p*q such that p*q+i*(p+q) is prime for i from 1 to n but not n+1.

4, 6, 15, 155, 35, 851, 34601, 474869, 15157931, 1467715961

Offset

0,1

Comments

If a(10) exists, then it is > 10^11. - Lucas A. Brown, Mar 21 2024

From David A. Corneth, Mar 24 2024: (Start)

a(10) <= 12948251561621.

a(11) <= 5670767031641.

If a(n) = p*q then (product of primes <= n) | (p + q). (End)

Links

Table of n, a(n) for n=0..9.

Lucas A. Brown, Python program.

Example

a(4) = 35 = 5*7 because 5+7 = 12 and 35+12 = 47, 35+2*12 = 59, 35+3*12 = 71, and 35+4*12 = 83 are prime but 35+5*12 = 95 is not, and 35 is the least semiprime that works.

Maple

V:= Array(0..8): V[0]:= 4: V[1]:= 6: count:= 2:
for n from 9 by 2 while count < 9 do
   if numtheory:-bigomega(n) = 2 then
     P:= numtheory:-factorset(n);
     if nops(P) = 2 then s:= P[1]+P[2] else s:= 2*P[1] fi;
     for i from 1 while isprime(n+i*s) do od:
     v:=i-1;
     if V[v] = 0 then V[v]:= n; count:= count+1; fi
   fi
od:
convert(V, list);

Mathematica

m[p_, q_] := Module[{i = 1}, While[PrimeQ[p*q + i*(p + q)], i++]; i]; seq[len_, nmax_] := Module[{s = Table[0, {len}], n = 1, c = 0, i, f}, While[c < len && n < nmax, f = FactorInteger[n]; If[f[[;; , 2]] == {2} || f[[;; , 2]] == {1, 1}, p = f[[1, 1]]; q = n/p; i = m[p, q]; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]]; n++]; s]; seq[8, 10^6] (* Amiram Eldar, Aug 28 2022 *)

Prog

(PARI)

Crossrefs

Cf. A001358, A034386, A356825.

Sequence in context: A305580 A191311 A086714 * A239323 A009463 A066260

Adjacent sequences: A356798 A356799 A356800 * A356802 A356803 A356804

Keyword

nonn,more,hard

Author

J. M. Bergot and Robert Israel, Aug 28 2022

Extensions

a(9) from Amiram Eldar, Aug 28 2022

Status

approved