A392914 Primes starting a run of at least 4 consecutive primes with binary Hamming distance 1.

449, 2213, 2593, 2657, 6883, 21377, 23057, 26083, 29009, 37313, 39041, 41161, 42689, 45329, 49253, 55889, 65537, 70177, 71249, 90017, 92369, 92849, 113761, 120929, 129893, 143249, 150217, 155081, 156749, 160481, 166273, 166289, 167009, 174241, 190577, 193937, 198929, 206177, 218401, 240257, 249857

Offset

1,1

Comments

Binary hamming distance is d(x,y) = A101080(x,y) and a prime p here followed by primes q, r, s has d(p,q)=1, d(q,r)=1, and d(r,s)=1.

These correspond to three consecutive 1s in sequence A205510.

Links

Table of n, a(n) for n=1..41.

Example

166273 is in the sequence because the 4 consecutive primes are:
  166273 = 101000100110000001_2
  166289 = 101000100110010001_2
  166297 = 101000100110011001_2
  166301 = 101000100110011101_2
and the successive binary Hamming distances are 1, 1, 1.
166297 is not in the sequence because the 4 consecutive primes are:
  166297 = 101000100110011001_2
  166301 = 101000100110011101_2
  166303 = 101000100110011111_2
  166319 = 101000100110101111_2
and the successive binary Hamming distances are 1, 1, 2.

Mathematica

Select[Partition[Prime[Range[26000]], 4, 1], AllTrue[DigitSum[BitXor @@@ Partition[#, 2, 1], 2], # == 1 &] &][[;; , 1]] (* Amiram Eldar, Jan 27 2026 *)

Prog

(Python)
from sympy import primerange
def hamming_quadruples(limit):
    primes = list(primerange(2, limit))
    for p1, p2, p3, p4 in zip(primes, primes[1:], primes[2:], primes[3:]):
        if (bin(p1 ^ p2).count('1') == 1 and
            bin(p2 ^ p3).count('1') == 1 and
            bin(p3 ^ p4).count('1') == 1):
            print(p1, end=", ")
hamming_quadruples(500000)

Crossrefs

Cf. A101080 (Hamming distance), A205510 (successive primes distance), A205511 (Lesser of Hamming twins), A205533 (Middle of Hamming triples), A007530 (Prime quadruples).

Sequence in context: A135073 A160291 A193255 * A283791 A036945 A201182

Adjacent sequences: A392911 A392912 A392913 * A392915 A392916 A392917

Keyword

nonn,base

Author

Jens Ahlström, Jan 27 2026

Status

approved