cp's OEIS Frontend

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.

%I A322035 #21 Jun 20 2025 20:23:38
%S A322035 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,
%T A322035 14,29,10,31,32,11,17,35,36,37,19,39,10,41,42,43,22,15,23,47,24,49,50,
%U A322035 17,13,53,27,55,28,57,29,59,20,61,31,63,64,65,22
%N 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.
%H A322035 Seiichi Manyama, <a href="/A322035/b322035.txt">Table of n, a(n) for n = 1..16384</a>
%e A322035 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.
%e A322035 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, ...
%p A322035 # This generates the terms starting at n=2:
%p A322035 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
%p A322035 f0:=[]; f1:=[]; f2:=[];
%p A322035 for n from 2 to 120 do
%p A322035 a:=0; b:=1; t1:=[P(n)];
%p A322035 for i from 1 to nops(t1) do b:=b/t1[i]; a:=a+b; od;
%p A322035 f0:=[op(f0),a]; f1:=[op(f1), numer(a)]; f2:=[op(f2),denom(a)]; od:
%p A322035 f0;    # s
%p A322035 f1;    # A322034
%p A322035 f2;    # A322035
%p A322035 f2-f1; # A322036
%t A322035 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 *)
%Y A322035 Cf. A006022, A027746, A322034, A322036.
%Y A322035 A017665/A017666 = sum of reciprocals of all divisors of n.
%K A322035 nonn,frac
%O A322035 1,2
%A A322035 _N. J. A. Sloane_ and _David James Sycamore_, Nov 28 2018