A335431 Numbers of the form q*(2^k), where q is one of the Mersenne primes (A000668) and k >= 0.

3, 6, 7, 12, 14, 24, 28, 31, 48, 56, 62, 96, 112, 124, 127, 192, 224, 248, 254, 384, 448, 496, 508, 768, 896, 992, 1016, 1536, 1792, 1984, 2032, 3072, 3584, 3968, 4064, 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

Offset: 1

Author

Antti Karttunen, Jun 28 2020

Keywords

• nonn

Comments

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

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

Numbers k for which A331410(k) = 1.

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

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.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

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

Row 1 of A335430.

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

Subsequence of A054784.

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

Programs

Mathematica

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 *)

PARI

A000265(n) = (n>>valuation(n,2));
isA000668(n) = (isprime(n)&&!bitand(n,1+n));
isA335431(n) = isA000668(A000265(n));

Formula

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

Sum_{n>=1} 1/a(n) = 2 * A173898 = 1.0329083578... - Amiram Eldar, Feb 18 2021