A335431 - OEIS

%I #34 Jul 28 2023 10:39:47

%S 3,6,7,12,14,24,28,31,48,56,62,96,112,124,127,192,224,248,254,384,448,

%T 496,508,768,896,992,1016,1536,1792,1984,2032,3072,3584,3968,4064,

%U 6144,7168,7936,8128,8191,12288,14336,15872,16256,16382,24576,28672,31744,32512,32764,49152,57344,63488,65024,65528,98304,114688,126976,130048,131056,131071

%N Numbers of the form q*(2^k), where q is one of the Mersenne primes (A000668) and k >= 0.

%C Numbers of the form 2^k * ((2^p)-1), where p is one of the primes in A000043, and k >= 0.

%C Numbers k such that A000265(k) is in A000668.

%C Numbers k for which A331410(k) = 1.

%C Numbers k that themselves are not powers of two, but for which A335876(k) = k+A052126(k) is [a power of 2].

%C Conjecture: This sequence gives all fixed points of map n -> A332214(n) and its inverse n -> A332215(n). See also notes in A029747 and in A163511.

%H Amiram Eldar, <a href="/A335431/b335431.txt">Table of n, a(n) for n = 1..10000</a>

%F A332214(a(n)) = A332215(a(n)) = a(n) for all n.

%F Sum_{n>=1} 1/a(n) = 2 * A173898 = 1.0329083578... - _Amiram Eldar_, Feb 18 2021

%t qs = 2^MersennePrimeExponent[Range[6]] - 1; max = qs[[-1]]; Reap[Do[n = 2^k*q; If[n <= max, Sow[n]], {k, 0, Log2[max]}, {q, qs}]][[2, 1]] // Union (* _Amiram Eldar_, Feb 18 2021 *)

%o (PARI)

%o A000265(n) = (n>>valuation(n,2));

%o isA000668(n) = (isprime(n)&&!bitand(n,1+n));

%o isA335431(n) = isA000668(A000265(n));

%Y Cf. A000043, A000396 (even terms form a subsequence), A000668 (primes present), A335882, A341622.

%Y Row 1 of A335430.

%Y Positions of 1's in A331410, in A364260, and in A364251 (characteristic function).

%Y Subsequence of A054784.

%Y Cf. also A029747, A163511, A173898, A332214, A332215, A334101, A335876.

%K nonn

%O 1,1

%A _Antti Karttunen_, Jun 28 2020