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A308732 Primes p such that the smallest possible number of 1's in binary representation of a multiple of p equals 3. 1
7, 23, 47, 71, 73, 79, 103, 151, 167, 191, 199, 239, 263, 271, 311, 337, 359, 367, 383, 439, 463, 479, 487, 503, 599, 607, 631, 647, 719, 727, 743, 751, 823, 839, 863, 887, 919, 937, 967, 983, 991, 1031, 1039, 1063, 1087, 1151, 1223, 1231, 1279, 1289, 1303 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
LINKS
C. Elsholtz, Almost all primes have a multiple of small Hamming weight, Bull. Aust. Math. Soc. 94 (2016), 224-235.
K. B. Stolarsky, Integers whose multiples have anomalous digital frequencies, Acta Arithmetica 38 (1980), 117-128.
MAPLE
filter:= proc(n) local S, r, j;
if not isprime(n) then return false fi;
r:= numtheory:-order(2, n);
if r::even then return false fi;
S:= {seq(2 &^ j mod n, j=1..r)};
S intersect map(t -> -t-1 mod n, S) <> {}
end proc:
select(filter, [seq(i, i=3..2000, 2)]); # Robert Israel, Jun 23 2019
CROSSREFS
Cf. A014662, which enumerates the same sequence for two 1's instead of three.
Sequence in context: A000353 A097149 A185007 * A139035 A002146 A336092
KEYWORD
nonn,base
AUTHOR
Jeffrey Shallit, Jun 20 2019
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)