A383891 a(n) is the length of chunks of the prime number sequence such that each chunk's sum of reciprocals is no less than 1/n, chunks being consecutive and of minimal length, for n>=2.

1, 1, 2, 3, 5, 8, 13, 22, 36, 60, 100, 168, 284, 482, 819, 1397, 2389, 4096, 7044, 12137, 20956, 36259, 62843, 109101, 189710

Offset

2,3

Comments

It is curious that the initial terms look like the Fibonacci sequence (A000045) or the 1-dimension sandpile sequence (A186085).

What limit is a(n)/a(n-1) approaching? - Bill McEachen, Sep 13 2025

Links

Table of n, a(n) for n=2..26.

Example

1/2 <= 1/2, so a(2) = 1.
1/3 <= 1/3, so a(3) = 1.
1/4 <= 1/5 + 1/7, so a(4) = 2.
1/5 <= 1/11 + 1/13 + 1/17, so a(5) = 3.
1/6 <= 1/19 + 1/23 + 1/29 + 1/31 + 1/37, so a(6) = 5.

Prog

(Haskell)
import Data.Numbers.Primes (primes)
import Data.Ratio ((%))
list = map length (splitBySum (reciprocals [2 ..]) (reciprocals primes))
reciprocals = map (1 %)
splitBySum (x : xs) a = a1 : splitBySum xs a2 where
  (a1, a2) = splitAt (len + 1) a
  len = length (takeWhile (< x) (scanl1 (+) a))
(PARI) lista(limit=10^5)={my(L=List(), s=0, m=1/2, c=0); forprime(p=2, limit, c++; s+=1/p; if(s>=m, listput(L, c); c=0; s=0; m=1/(#L+2))); Vec(L)} \\ Andrew Howroyd, Sep 13 2025

Crossrefs

Cf. A000040.

Sequence in context: A245271 A206743 A186085 * A018151 A227374 A124429

Adjacent sequences: A383888 A383889 A383890 * A383892 A383893 A383894

Keyword

nonn,more

Author

Xiaoliang Zhang, May 13 2025

Extensions

a(24)-a(26) from Bill McEachen, Sep 13 2025

Status

approved