OFFSET
1,4
COMMENTS
Note that s < 1 for all n (compare A322036). This follows easily by induction, since when we increase n by multiplying it by a new (not-smaller) prime, we increase s by less than 1-s.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
EXAMPLE
If n=12 we get the prime factors 2,2,3, and s = 1/2 + 1/4 + 1/12 = 5/6. So a(12) = 5.
The fractions s for n >= 2 are 1/2, 1/3, 3/4, 1/5, 2/3, 1/7, 7/8, 4/9, 3/5, 1/11, 5/6, 1/13, 4/7, 2/5, 15/16, 1/17, 13/18, 1/19, 4/5, 8/21, ...
MAPLE
# This generates the terms starting at n=2:
P:=proc(n) local FM: FM:=ifactors(n)[2]: seq(seq(FM[j][1], k=1..FM[j][2]), j=1..nops(FM)) end: # A027746
f0:=[]; f1:=[]; f2:=[];
for n from 2 to 120 do
a:=0; b:=1; t1:=[P(n)];
for i from 1 to nops(t1) do b:=b/t1[i]; a:=a+b; od;
f0:=[op(f0), a]; f1:=[op(f1), numer(a)]; f2:=[op(f2), denom(a)]; od:
f0; # s
f1; # A322034
f2; # A322035
f2-f1; # A322036
PROG
(PARI) A322034(n) = if(1==n, 0, my(f=factor(n), pm=1, s=0); for(i=1, #f~, while(f[i, 2], pm *= f[i, 1]; f[i, 2]--; s += 1/pm)); numerator(s)); \\ Antti Karttunen, Feb 28 2019
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
N. J. A. Sloane and David James Sycamore, Nov 28 2018
STATUS
approved