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

%I A056652 #26 Sep 28 2020 01:03:51
%S A056652 3,7,9,21,27,31,49,63,81,93,127,147,189,217,243,279,343,381,441,567,
%T A056652 651,729,837,889,961,1029,1143,1323,1519,1701,1953,2187,2401,2511,
%U A056652 2667,2883,3087,3429,3937,3969,4557,5103,5859,6223,6561,6727,7203,7533,8001,8191,8649,9261
%N A056652 Integers > 1 whose prime divisors are all Mersenne primes (i.e., of the form (2^p - 1)).
%H A056652 Amiram Eldar, <a href="/A056652/b056652.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..471 from Michael De Vlieger)
%F A056652 Sum_{n>=1} 1/a(n) = - 1 + Product_{p in A000668} p/(p-1) = 0.82292512097260346512... - _Amiram Eldar_, Sep 27 2020
%e A056652 63 is included because the prime factorization of 63 is 3^2 * 7 = (2^2 -1)^2 *(2^3 -1).
%p A056652 isA000668 := proc(n)
%p A056652     if n in [   3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727] then
%p A056652         true;
%p A056652     else
%p A056652         false;
%p A056652     end if;
%p A056652 end proc:
%p A056652 isA056652 := proc(n)
%p A056652     local p;
%p A056652     for p in numtheory[factorset](n) do
%p A056652         if not isA000668(p) then
%p A056652             return false;
%p A056652         end if;
%p A056652     end do:
%p A056652     true ;
%p A056652 end proc:
%p A056652 for n from 2 to 1000 do
%p A056652     if isA056652(n) then
%p A056652         printf("%d,",n);
%p A056652     end if;
%p A056652 end do: # _R. J. Mathar_, Feb 19 2017
%t A056652 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 *)
%o A056652 (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
%Y A056652 Cf. A000668, A046528.
%K A056652 nonn
%O A056652 1,1
%A A056652 _Leroy Quet_, Aug 09 2000
%E A056652 Offset corrected and more terms added by _Michel Marcus_, Apr 25 2016