A209085 -id:A209085 - cp's OEIS frontend

A208982 Numbers n such that the next larger number with mutual Hamming distance 1 is prime.

1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 16, 17, 18, 19, 21, 22, 23, 27, 28, 29, 30, 36, 39, 40, 41, 42, 43, 45, 46, 52, 57, 58, 60, 63, 65, 66, 67, 69, 70, 71, 72, 75, 77, 78, 81, 82, 88, 95, 96, 99, 100, 101, 102, 105, 106, 108, 111, 112, 119, 123, 125, 126, 129, 130, 136, 137, 138, 147, 148, 149, 150

Offset: 1

Vladimir Shevelev, Mar 04 2012

If p is prime, then p-1 is in the sequence.

Using the prime number theorem in arithmetic progressions k*n+b with gcd(k,b)=1 and its uniformity over k<=exp(c*sqrt(log(x))), one can prove that the counting function of a(n)<=x is equivalent to 2*x/log(x), as x tends to infinity.

Cf. A205509, A205510, A205511, A205302, A205649, A205533, A122565, A206852, A206853, A206960, A209085.

isok(n) = my(nextn = n+1); while (hammingweight(bitxor(n, nextn)) != 1, nextn++); isprime(nextn); \\ Michel Marcus, Jul 01 2014

A209332 a(n) is the minimal positive number k such that n<+>k is prime or 0 if no such number exists (operation <+> defined in A206853).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 3, 5, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 0, 3, 1, 1, 1, 1, 2, 4, 0, 3, 2, 1, 0, 4, 1, 1, 1, 1, 1, 2, 1, 1, 0, 5, 0, 3, 2, 1, 0, 7, 2, 2, 1, 1, 2, 1, 0, 8, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 0, 7, 1, 2, 1, 1, 3, 2, 1, 1, 0, 3, 0, 4, 3

Offset: 1

Vladimir Shevelev, Mar 06 2012

Numbers n for which a(n) = 1 form sequence A208982.
a(n) = 0 for n = 25, 33, 37, 47,... (A209333).
A simple sufficient condition for a(n) = 0 (which is proved by induction) is that n<+>k is not prime up to the moment that n<+>k is even and n<+>(k+1)-n<+>k = 2^t, where t >= m+1 and m defined by the condition 2^m <= n < 2^(m+1).
Conjecture: for even n, a(n) > 0.

Cf. A205509, A205510, A205511, A205302, A205649, A205533, A122565, A206852, A206853, A206960, A209085, A208982, A209085, A209333.

cp's OEIS Frontend

A208982 Numbers n such that the next larger number with mutual Hamming distance 1 is prime.

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