A322035 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) = denominator of s. a(1)=1 by convention.

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 6, 13, 7, 5, 16, 17, 18, 19, 5, 21, 11, 23, 12, 25, 13, 27, 14, 29, 10, 31, 32, 11, 17, 35, 36, 37, 19, 39, 10, 41, 42, 43, 22, 15, 23, 47, 24, 49, 50, 17, 13, 53, 27, 55, 28, 57, 29, 59, 20, 61, 31, 63, 64, 65, 22

Offset: 1

Author

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

Keywords

• nonn
• frac

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) = 6.
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

• Seiichi Manyama, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A006022, A027746, A322034, 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]]; Table[Denominator@ Total@ Table[1/Times @@ #[[;; i]], {i, Length[#]}] &@ f[n], {n, 120}] (* Michael De Vlieger, Jun 20 2025 *)