|
| |
|
|
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.
|
|
0
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
|
| |
|
|
