A056652 Integers > 1 whose prime divisors are all Mersenne primes (i.e., of the form (2^p - 1)).

3, 7, 9, 21, 27, 31, 49, 63, 81, 93, 127, 147, 189, 217, 243, 279, 343, 381, 441, 567, 651, 729, 837, 889, 961, 1029, 1143, 1323, 1519, 1701, 1953, 2187, 2401, 2511, 2667, 2883, 3087, 3429, 3937, 3969, 4557, 5103, 5859, 6223, 6561, 6727, 7203, 7533, 8001, 8191, 8649, 9261

Offset: 1

Author

Leroy Quet, Aug 09 2000

Keywords

• nonn

Examples

			63 is included because the prime factorization of 63 is 3^2 * 7 = (2^2 -1)^2 *(2^3 -1).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..471 from Michael De Vlieger)

Crossrefs

Cf. A000668, A046528.

Programs

Maple

isA000668 := proc(n)
    if n in [   3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727] then
        true;
    else
        false;
    end if;
end proc:
isA056652 := proc(n)
    local p;
    for p in numtheory[factorset](n) do
        if not isA000668(p) then
            return false;
        end if;
    end do:
    true ;
end proc:
for n from 2 to 1000 do
    if isA056652(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Feb 19 2017

Mathematica

Block[{nn = 10^4, s}, s = TakeWhile[Select[2^Prime@ Range@ 8 - 1, PrimeQ], # <= nn &]; Select[Range@ nn, AllTrue[FactorInteger[#][[All, 1]], MemberQ[s, #] &] &]] (* Michael De Vlieger, Sep 03 2017 *)

PARI

isok(n) = {if (n==1, return (0)); my(f = factor(n)); for (k=1, #f~, if (! ((q=ispower(f[k, 1]+1,,&e)) && isprime(q) && (e==2)), return(0));); 1;} \\ Michel Marcus, Apr 25 2016

Formula

Sum_{n>=1} 1/a(n) = - 1 + Product_{p in A000668} p/(p-1) = 0.82292512097260346512... - Amiram Eldar, Sep 27 2020

Extensions

Offset corrected and more terms added by Michel Marcus, Apr 25 2016