A350230 Numbers m such that for all factorizations m=a*b with positive integers (a, b), a*b+a+b is a prime.

1, 2, 3, 5, 6, 11, 14, 15, 21, 23, 26, 29, 30, 33, 35, 41, 51, 53, 65, 74, 83, 86, 89, 111, 113, 131, 141, 155, 158, 173, 179, 186, 191, 194, 209, 215, 221, 230, 231, 233, 239, 251, 254, 278, 281, 293, 321, 323, 326, 329, 341, 345, 359, 371, 398, 413, 419, 426

Offset: 1

Author

Michel Lagneau, Dec 21 2021

Keywords

• nonn

Comments

From Marius A. Burtea, Dec 26 2021: (Start)

If p is a prime number in A005384 (Sophie Germain prime), then it is a term. Indeed, p = p*1 and p + p + 1 = 2*p + 1 is prime.

All terms (m >= 2) are squarefree numbers (A005117). Indeed, if m = p^2*k, p >= 2, k >= 1, then m = p*(p*k) and p^2*k + p + p*k = p*(p*k + 1 + k ) is not prime. (End).

Examples

			1 is in the sequence because 1 = 1*1 and 1*1 + 1 + 1 = 3 is prime;
30 is in the sequence because A038548(30) = 4 has 4 factorizations:
30 = 1*30 = 2*15 = 3*10 = 5*6 and
30 + 1 + 30 = 61 is prime;
30 + 2 + 15 = 47 is prime;
30 + 3 + 10 = 43 is prime;
30 + 5 + 6 = 41 is prime.
		

Crossrefs

Cf. A005117 (squarefree numbers), A005384 (Sophie Germain primes), A038548, A080715.

Programs

Magma

[n:n in [1..450]|forall{d: d in Divisors(n)| IsPrime(n+d+n div d)}]; // Marius A. Burtea, Dec 26 2021

Maple

A350230 := proc(n)
    local a,b ;
    for a in numtheory[divisors](n) do
        b := n/a ;
        if not isprime(a*b+b+a) then
            return false;
        end if;
    end do:
    true ;
end proc:
for n from 1 to 500 do
    if isA350230(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jan 24 2022

Mathematica

t={};Do[ds=Divisors[n];If[EvenQ[Length[ds]],ok=True;k=1;While[k<=Length[ds]/2&&(ok=PrimeQ[ds[[k]]*ds[[-k]]+ds[[k]]+ds[[-k]]]),k++];If[ok,AppendTo[t,n]]],{n,2,4000}];t

PARI

isok(m) = sumdiv(m, d, isprime(m+d+m/d)) == numdiv(m); \\ Michel Marcus, Dec 25 2021

Python

from sympy import divisors, isprime
def ok(n):
    divs = divisors(n)
    if n == 0: return False
    return all(isprime(a*b+a+b) for a, b in ((d, n//d) for d in divs))
print([k for k in range(427) if ok(k)]) # Michael S. Branicky, Dec 21 2021