cp's OEIS Frontend

A308732 Primes p such that the smallest possible number of 1's in binary representation of a multiple of p equals 3.

%I A308732 #20 Dec 19 2024 12:19:45
%S A308732 7,23,47,71,73,79,103,151,167,191,199,239,263,271,311,337,359,367,383,
%T A308732 439,463,479,487,503,599,607,631,647,719,727,743,751,823,839,863,887,
%U A308732 919,937,967,983,991,1031,1039,1063,1087,1151,1223,1231,1279,1289,1303
%N A308732 Primes p such that the smallest possible number of 1's in binary representation of a multiple of p equals 3.
%C A308732 The first few corresponding multipliers that give three 1's are (for the numbers listed above) are 1, 3, 11, 119, 1, 13, 5, 7, 791, 87839, 247, 17970575, 3987, 8048111, 7, 49, 23, 2995944847, 5607007, 7, 2319663.
%H A308732 Robert Israel, <a href="/A308732/b308732.txt">Table of n, a(n) for n = 1..10000</a>
%H A308732 Christian Elsholtz, <a href="https://doi.org/10.1017/S000497271600023X">Almost all primes have a multiple of small Hamming weight</a>, Bull. Aust. Math. Soc. 94 (2016), 224-235.
%H A308732 Eugen J. Ionascu, Florian Luca, and Thomas Merino, <a href="https://arxiv.org/abs/2412.10839">On the average value of the minimal Hamming multiple</a>, arXiv:2412.10839 [math.NT], 2024. See pp. 4, 17.
%H A308732 Kenneth B. Stolarsky, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa38/aa3825.pdf">Integers whose multiples have anomalous digital frequencies</a>, Acta Arithmetica 38 (1980), 117-128.
%p A308732 filter:= proc(n) local S, r, j;
%p A308732   if not isprime(n) then return false fi;
%p A308732   r:= numtheory:-order(2,n);
%p A308732   if r::even then return false fi;
%p A308732   S:= {seq(2 &^ j mod n, j=1..r)};
%p A308732   S intersect map(t -> -t-1 mod n, S) <> {}
%p A308732 end proc:
%p A308732 select(filter, [seq(i,i=3..2000, 2)]); # _Robert Israel_, Jun 23 2019
%Y A308732 Cf. A014662, which enumerates the same sequence for two 1's instead of three.
%K A308732 nonn,base
%O A308732 1,1
%A A308732 _Jeffrey Shallit_, Jun 20 2019