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 A332103 Numbers not of the form floor(p/4) + 1, where p is a prime. 0

%I #8 Oct 21 2020 22:40:55

%S 0,7,9,13,22,24,30,31,34,36,37,39,43,47,51,52,54,55,62,64,67,69,72,73,

%T 75,76,81,82,84,86,91,93,97,99,102,104,107,112,114,118,119,121,124,

%U 127,129,130,132,133,134,135,138,139,142,144,146,148

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

%C 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).

%e 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.

%t m = 150; Complement[Range[0, m], Floor[Select[Range[4*m], PrimeQ]/4] + 1] (* _Amiram Eldar_, Oct 21 2020 *)

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

%Y Cf. A000040, A001348.

%K nonn,easy

%O 1,2

%A _M. F. Hasler_, Oct 20 2020

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