A333877 -id:A333877 - cp's OEIS frontend

A333876 a(n) is the largest prime 2^(n-1) <= p < 2^n minimizing the Hamming weight of all primes in this interval.

2, 5, 13, 17, 41, 97, 193, 257, 769, 1153, 2113, 4129, 12289, 18433, 40961, 65537, 163841, 270337, 786433, 1179649, 2101249, 4194433, 8650753, 16777729, 50332673, 69206017, 167772161, 270532609, 537133057, 1107296257, 3221225473, 6442713089, 8858370049

Offset: 2

Hugo Pfoertner, Apr 08 2020

nonn

Chai Wah Wu, Table of n, a(n) for n = 2..1000

Cf. A066195, A091936, A333877, A333878.

for(n=2, 10, my(hmin=n+n,pmax); forprime(p=2^(n-1), 2^n, my(h=hammingweight(p)); if(h<=hmin,pmax=p;hmin=h)); print1(pmax,", "))

from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def A333876(n):
    for i in range(n):
        q = 2**n-1
        for d in multiset_permutations('0'*i+'1'*(n-1-i)):
            p = q-int(''.join(d),2)
            if isprime(p):
                return p # Chai Wah Wu, Apr 08 2020

A333879 a(n) is the number of primes 2^(n-1) <= p < 2^n with maximal Hamming weight A091937(n-1) in this interval.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 4, 3, 1, 5, 1, 4, 18, 3, 1, 8, 1, 11, 4, 5, 31, 7, 1, 2, 33, 1, 5, 4, 1, 7, 5, 1, 1, 9, 55, 6, 71, 7, 1, 6, 69, 4, 7, 2, 1, 10, 3, 3, 1, 2, 1, 6, 95, 4, 3, 61, 1, 8, 1, 3, 110, 3, 1, 8, 1, 2, 2, 3, 98, 9, 1, 5, 2, 5, 8, 3, 125, 10, 3, 81, 2

Offset: 2

Hugo Pfoertner, Apr 08 2020

nonn

Giovanni Resta, Table of n, a(n) for n = 2..1000

Cf. A091937, A091938, A333877, A333878.

a[2]=1; a[n_] := Block[{c = 0, nb = 0}, While[c == 0, c = Count[2^n - 1 - Total /@ Subsets[2^Range[n - 2], {nb}], x_ /; PrimeQ[x]]; nb++]; c]; a /@ Range[2, 85] (* Giovanni Resta, Apr 09 2020 *)

Terms a(35) and beyond from Giovanni Resta, Apr 09 2020

cp's OEIS Frontend

A333876 a(n) is the largest prime 2^(n-1) <= p < 2^n minimizing the Hamming weight of all primes in this interval.

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