A370305 Numbers k such that the distance from exp(k) to the closest average of two consecutive primes is less than 1.

1, 3, 16, 61, 74, 91, 113, 1441, 1566, 2170, 2499

Offset

1,2

Comments

Explicitly, abs( e^k - (prevprime(e^k)+nextprime(e^k))/2 ) < 1.

For k>1, the formula (prevprime(e^k)+nextprime(e^k))/2 either gives floor(e^k), for k = 61, 74, 2170, ..., or gives ceiling(e^k), for k = 3, 16, 91, 113, 1441, 1566, 2499, ... This partitions {a(n)}\{1} into two subsequences each of which can be conjectured to have relative density 1/2.

In cases k = 16, 61, 113, 2499, ... the distance is actually less than 0.5. Then the formula (prevprime(e^k)+nextprime(e^k))/2 yields round(e^k), the nearest integer to e^k.

Links

Table of n, a(n) for n=1..11.

Example

For k=16, e^16 is about 8886110.52. The next prime is 8886113, and the previous prime is 8886109, and their average 8886111 is at a distance of about 0.48 away from e^16.

Prog

(PARI) default(realprecision, 2000); for(k=1, +oo, r=exp(k); abs(r-(precprime(r)+nextprime(r))/2)<1&&print1(k, ", "))

Crossrefs

Cf. A037028, A040016, A050808, A059303, A074496.

Sequence in context: A073999 A259056 A155160 * A355645 A323941 A267036

Adjacent sequences: A370302 A370303 A370304 * A370306 A370307 A370308

Keyword

nonn,hard,more

Author

Jeppe Stig Nielsen, Feb 14 2024

Status

approved