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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.

%I A383891 #11 May 19 2025 17:47:16
%S A383891 1,1,2,3,5,8,13,22,36,60,100,168,284,482,819,1397,2389,4096,7044,
%T A383891 12137,20956,36259
%N 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.
%C A383891 It is curious that the initial terms look like the Fibonacci sequence (A000045) or the 1-dimension sandpile sequence (A186085).
%e A383891 1/2 <= 1/2, so a(2) = 1.
%e A383891 1/3 <= 1/3, so a(3) = 1.
%e A383891 1/4 <= 1/5 + 1/7, so a(4) = 2.
%e A383891 1/5 <= 1/11 + 1/13 + 1/17, so a(5) = 3.
%e A383891 1/6 <= 1/19 + 1/23 + 1/29 + 1/31 + 1/37, so a(6) = 5.
%o A383891 (Haskell)
%o A383891 import Data.Numbers.Primes (primes)
%o A383891 import Data.Ratio ((%))
%o A383891 list = map length (splitBySum (reciprocals [2 ..]) (reciprocals primes))
%o A383891 reciprocals = map (1 %)
%o A383891 splitBySum (x : xs) a = a1 : splitBySum xs a2 where
%o A383891   (a1, a2) = splitAt (len + 1) a
%o A383891   len = length (takeWhile (< x) (scanl1 (+) a))
%Y A383891 Cf. A000040.
%K A383891 nonn,more
%O A383891 2,3
%A A383891 _Xiaoliang Zhang_, May 13 2025