cp's OEIS Frontend

A339061 Least integer j such that H(k+j)>=n+1, where k is the least integer to satisfy H(k)>=n, and H(k) is the sum of the first k terms of the harmonic series.

%I A339061 #29 Dec 15 2022 14:24:39
%S A339061 1,3,7,20,52,144,389,1058,2876,7817,21250,57763,157017,426817,1160207,
%T A339061 3153770,8572836,23303385,63345169,172190019,468061001,1272321714,
%U A339061 3458528995,9401256521,25555264765,69466411833,188829284972,513291214021
%N A339061 Least integer j such that H(k+j)>=n+1, where k is the least integer to satisfy H(k)>=n, and H(k) is the sum of the first k terms of the harmonic series.
%F A339061 a(n) ~ (e-1)*e^(n-gamma), where e is Euler's number and gamma is the Euler-Mascheroni constant.
%F A339061 Conjecture: a(n) = floor(1/2 + e^(n-gamma+1)) - floor(1/2 + e^(n-gamma)) for n > 1 where e is Euler's number and gamma is the Euler-Mascheroni constant. - _Adam Hugill_, Nov 06 2022
%e A339061 Define H(0)=0, H(k) = Sum_{i=1..k} 1/i for k=1,2,3,...
%e A339061 a(0)=1: To reach n+1 from n=0 requires 1 additional term of the harmonic partial sum: H(0+1) = H(0) + 1/1 = H(1) = 1.
%e A339061 a(1)=3: To reach n+1 from n=1 requires 3 additional terms of the harmonic partial sum: H(1+3) = H(1) + 1/(1+1) + 1/(1+2) + 1/(1+3) = H(4) = 2.08333....
%e A339061 a(2)=7: To reach n+1 from n=2 requires 7 additional terms of the harmonic partial sum: H(4+7) = H(4) + 1/(4+1) + 1/(4+2) + ... + 1/(4+6) + 1/(4+7) = H(11) = 3.01987....
%e A339061 a(3)=20: To reach n+1 from n=3 requires 20 additional terms of the harmonic partial sum: H(11+20) = H(11) + 1/(11+1) + 1/(11+2) + ... + 1/(11+19) + 1/(11+20) = H(31) = 4.02724....
%o A339061 (R)
%o A339061 #set size of search space
%o A339061 Max=10000000
%o A339061 #initialize sequence to empty
%o A339061 seq=vector(length=0)
%o A339061 #initialize partial sum to 0
%o A339061 partialsum=0
%o A339061 k=1
%o A339061 n=1
%o A339061 for(i in 1:Max){
%o A339061    partialsum=partialsum+1/i
%o A339061    if(partialsum>=n){
%o A339061       seq=c(seq, k)
%o A339061       k=0
%o A339061       n=n+1
%o A339061    }
%o A339061    k=k+1
%o A339061 }
%o A339061 #print sequence numbers below Max
%o A339061 seq
%Y A339061 First differences of A004080.
%Y A339061 Cf. A001113 (e), A001620 (gamma).
%Y A339061 Cf. A001008/A002805 (harmonic numbers).
%Y A339061 Some sequences in the same spirit as this: A331028, A002387, A004080.
%K A339061 nonn
%O A339061 0,2
%A A339061 _Matthew J. Bloomfield_, Dec 21 2020