A322034 Let p1 <= p2 <= ... <= pk be the prime factors of n, with repetition; let s = 1/p1 + 1/(p1*p2) + 1/(p1*p2*p3) + ... + 1/(p1*p2*...*pk); a(n) = numerator of s. a(1)=0 by convention.

0, 1, 1, 3, 1, 2, 1, 7, 4, 3, 1, 5, 1, 4, 2, 15, 1, 13, 1, 4, 8, 6, 1, 11, 6, 7, 13, 11, 1, 7, 1, 31, 4, 9, 8, 31, 1, 10, 14, 9, 1, 29, 1, 17, 7, 12, 1, 23, 8, 31, 6, 10, 1, 20, 12, 25, 20, 15, 1, 17, 1, 16, 29, 63, 14, 15, 1, 13, 8, 43, 1, 67

Offset: 1

Author

N. J. A. Sloane and David James Sycamore, Nov 28 2018

Keywords

• nonn
• frac

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.

Examples

			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, ...
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Crossrefs

Cf. A006022, A027746, A322035, A322036.

A017665/A017666 = sum of reciprocals of all divisors of n.

Programs

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

Mathematica

f[x_] := Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[x]]; {0}~Join~Table[Numerator@ Total@ Table[1/Times @@ #[[;; i]], {i, Length[#]}] &@ f[n], {n, 2, 72}] (* Michael De Vlieger, Jun 20 2025 *)

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