A335302 a(n) = prime(n+1) mod (2^k) where k is the least positive integer such that floor(prime(n)/(2^k)) = floor(prime(n+1)/(2^k)) and prime(n) denotes the n-th prime number.

1, 5, 3, 11, 5, 17, 3, 7, 13, 3, 37, 9, 3, 7, 21, 11, 5, 67, 7, 9, 7, 19, 9, 33, 5, 3, 11, 5, 17, 15, 131, 9, 3, 21, 3, 13, 35, 7, 13, 19, 5, 15, 65, 5, 3, 19, 15, 35, 5, 9, 7, 17, 11, 257, 7, 13, 3, 21, 9, 3, 37, 19, 7, 9, 5, 75, 17, 11, 5, 33, 7, 15, 21, 11

Offset

1,2

Comments

In other words, the binary representation of a(n) is the smallest suffix to be overlaid on the binary representation of the n-th prime number to obtain that of the next prime number.

This sequence has similarities with A006519; here we consider consecutive prime numbers, there consecutive nonnegative integers.

There are no two consecutive equal terms.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Formula

a(n) <= prime(n+1) with equality iff prime(n+1) belongs to A014210.

Example

The first terms, alongside the binary representations of a(n) and of prime(n+1), are:
  n   a(n)  bin(a(n))  bin(prime(n+1))
  --  ----  ---------  ---------------
   0   N/A        N/A               10
   1     1          1               11
   2     5        101              101
   3     3         11              111
   4    11       1011             1011
   5     5        101             1101
   6    17      10001            10001
   7     3         11            10011
   8     7        111            10111
   9    13       1101            11101
  10     3         11            11111

Prog

(PARI) { base=2; p=2; forprime (q=p+1, 379, for (k=0, oo, m=base^k; if (q\m == p\m, print1 (q%m", "); p=q; break))) }

Crossrefs

Cf. A006519, A014210, A335301 (decimal variant).

Sequence in context: A049829 A258333 A137613 * A259650 A165670 A221869

Adjacent sequences: A335299 A335300 A335301 * A335303 A335304 A335305

Keyword

nonn,base

Author

Rémy Sigrist, May 31 2020

Status

approved