A332103 Numbers not of the form floor(p/4) + 1, where p is a prime.

0, 7, 9, 13, 22, 24, 30, 31, 34, 36, 37, 39, 43, 47, 51, 52, 54, 55, 62, 64, 67, 69, 72, 73, 75, 76, 81, 82, 84, 86, 91, 93, 97, 99, 102, 104, 107, 112, 114, 118, 119, 121, 124, 127, 129, 130, 132, 133, 134, 135, 138, 139, 142, 144, 146, 148

Offset

1,2

Comments

Also, numbers that are not the length, in hexadecimal digits, of a Mersenne number 2^p - 1 (A001348). This is the main motivation for considering this sequence. Indeed, base 16 can be considered as quite natural in relation with Mersenne numbers which are, except for the initial 3, all of the form 1F...F or 7F...F when written in base 16. Since a hexadecimal digit corresponds to four bits and small primes are relatively dense, most small positive integers are the length of some Mersenne number in base 16. Here we list the "exceptions" (which of course become less rare as the size of the primes and therewith the mean gap between them increase).

Links

Table of n, a(n) for n=1..56.

Example

Dividing the primes A000040 = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, ...} by 4, we get for the integer parts (without repetitions) {0, 1, 2, 3, 4, 5, 7, 9, ...}, where the first integers missing are 6 and 8. Increasing these by one we get the first nonzero terms of this sequence.

Mathematica

m = 150; Complement[Range[0, m], Floor[Select[Range[4*m], PrimeQ]/4] + 1] (* Amiram Eldar, Oct 21 2020 *)

Prog

(PARI) select( {is(n)=!#primes([n*4-3, n*4])}, [0..499])

Crossrefs

Cf. A000040, A001348.

Sequence in context: A297063 A185720 A032487 * A160777 A365986 A063189

Adjacent sequences: A332100 A332101 A332102 * A332104 A332105 A332106

Keyword

nonn,easy

Author

M. F. Hasler, Oct 20 2020

Status

approved