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.

1, 3, 7, 20, 52, 144, 389, 1058, 2876, 7817, 21250, 57763, 157017, 426817, 1160207, 3153770, 8572836, 23303385, 63345169, 172190019, 468061001, 1272321714, 3458528995, 9401256521, 25555264765, 69466411833, 188829284972, 513291214021

Offset

0,2

Links

Table of n, a(n) for n=0..27.

Formula

a(n) ~ (e-1)*e^(n-gamma), where e is Euler's number and gamma is the Euler-Mascheroni constant.

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

Example

Define H(0)=0, H(k) = Sum_{i=1..k} 1/i for k=1,2,3,...
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.
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....
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....
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....

Prog

(R)
#set size of search space
Max=10000000
#initialize sequence to empty
seq=vector(length=0)
#initialize partial sum to 0
partialsum=0
k=1
n=1
for(i in 1:Max){
   partialsum=partialsum+1/i
   if(partialsum>=n){
      seq=c(seq, k)
      k=0
      n=n+1
   }
   k=k+1
}
#print sequence numbers below Max
seq

Crossrefs

First differences of A004080.

Cf. A001113 (e), A001620 (gamma).

Cf. A001008/A002805 (harmonic numbers).

Some sequences in the same spirit as this: A331028, A002387, A004080.

Sequence in context: A230352 A271894 A066315 * A058499 A003097 A109220

Adjacent sequences: A339058 A339059 A339060 * A339062 A339063 A339064

Keyword

nonn

Author

Matthew J. Bloomfield, Dec 21 2020

Status

approved