cp's OEIS Frontend

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.

%I A322034 #37 Jun 20 2025 20:23:47
%S A322034 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,
%T A322034 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,
%U A322034 1,17,1,16,29,63,14,15,1,13,8,43,1,67
%N 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.
%C A322034 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.
%H A322034 Antti Karttunen, <a href="/A322034/b322034.txt">Table of n, a(n) for n = 1..16384</a>
%H A322034 Antti Karttunen, <a href="/A322034/a322034.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%e A322034 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.
%e A322034 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 A322034 # This generates the terms starting at n=2:
%p A322034 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 A322034 f0:=[]; f1:=[]; f2:=[];
%p A322034 for n from 2 to 120 do
%p A322034 a:=0; b:=1; t1:=[P(n)];
%p A322034 for i from 1 to nops(t1) do b:=b/t1[i]; a:=a+b; od;
%p A322034 f0:=[op(f0),a]; f1:=[op(f1), numer(a)]; f2:=[op(f2),denom(a)]; od:
%p A322034 f0;    # s
%p A322034 f1;    # A322034
%p A322034 f2;    # A322035
%p A322034 f2-f1; # A322036
%t A322034 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 *)
%o A322034 (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
%Y A322034 Cf. A006022, A027746, A322035, A322036.
%Y A322034 A017665/A017666 = sum of reciprocals of all divisors of n.
%K A322034 nonn,frac
%O A322034 1,4
%A A322034 _N. J. A. Sloane_ and _David James Sycamore_, Nov 28 2018