A335889 - OEIS

%I #29 Jul 20 2020 08:32:57

%S 1,2,0,3,1,0,0,3,1,0,0,2,0,3,2,1,0,0,0,3,0,2,1,0,1,1,2,1,0,0,4,0,0,0,

%T 2,0,2,3,3

%N a(n) is the number of Mersenne primes between consecutive perfect numbers.

%H Chris K. Caldwell and G. L. Honaker, Jr., <a href="https://primes.utm.edu/curios/page.php?curio_id=38700">Prime Curio for 756839</a>

%e a(1) = 1 because there is exactly 1 Mersenne prime (7) between the first and second perfect numbers (6 and 28).

%e a(4) = 3 because there are exactly 3 Mersenne primes (8191, 131071, 524287) between the fourth and fifth perfect numbers (8128 and 33550336).

%t p = MersennePrimeExponent @ Range[47];mer[p_] := 2^p - 1; perf[p_] := mer[p] * 2^(p - 1); mers = mer /@ p; perfs = Select[perf /@ p, # < mers[[-1]] &]; BinCounts[mers, {perfs}] (* _Amiram Eldar_, Jun 29 2020 *)

%Y Cf. A000043, A000396, A000668.

%K nonn,more

%O 1,2

%A _G. L. Honaker, Jr._, Jun 28 2020

%E a(5)-a(13) from _Metin Sariyar_, Jun 28 2020

%E a(14)-a(16) and a(20)-a(39) from _Metin Sariyar_, Jun 29 2020

%E a(17)-a(19) from _Amiram Eldar_, Jun 29 2020