cp's OEIS Frontend

A286385 a(n) = A003961(n) - A000203(n).

%I A286385 #45 Dec 21 2023 10:22:58
%S A286385 0,0,1,2,1,3,3,12,12,3,1,17,3,9,11,50,1,36,3,21,23,3,5,75,18,9,85,43,
%T A286385 1,33,5,180,17,3,29,134,3,9,29,99,1,69,3,33,97,15,5,281,64,54,23,55,5,
%U A286385 255,19,177,35,3,1,147,5,15,171,602,35,51,3,45,49,87,1,480,5,9,121,67,47,87,3,381,504,3,5,271,25,9,35,171,7,291,75,93,57,15,41,963
%N A286385 a(n) = A003961(n) - A000203(n).
%C A286385 Are all terms nonnegative? This question is equivalent to the question posed in A285705.
%C A286385 From _Antti Karttunen_, Aug 05 2020: (Start)
%C A286385 The answer to the above question is yes. Because both A000203 and A003961 are multiplicative sequences, it suffices to prove that for any prime p, and e >= 1, q^e >= sigma(p^e) = ((p^(1+e))-1) / (p-1), where q = A151800(p), i.e., the next larger prime after p. If p is a lesser twin prime, then q = p+2 (and this difference can't be less than 2, apart from case p=2), and it is easy to see that (n+2)^e > ((n^(e+1)) - 1) / (n-1), for all n >= 2, e >= 1.
%C A286385 See comments in A326042.
%C A286385 (End)
%C A286385 This is the inverse Möbius transform of A337549, from which it is even easier to see that all terms are nonnegative. - _Antti Karttunen_, Sep 22 2020
%H A286385 Antti Karttunen, <a href="/A286385/b286385.txt">Table of n, a(n) for n = 1..16383</a>
%F A286385 a(n) = A285705(A048673(n)) - 1 = 2*A048673(n) - A000203(n) - 1.
%F A286385 a(n) = A336852(n) - A336851(n). - _Antti Karttunen_, Aug 05 2020
%F A286385 a(n) = Sum_{d|n} A337549(d). - _Antti Karttunen_, Sep 22 2020
%F A286385 Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2-p)/(p^2-q(p))) - Pi^2/12 = 1.24152934..., where q(p) = nextprime(p) (A151800). - _Amiram Eldar_, Dec 21 2023
%t A286385 Array[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] - DivisorSigma[1, #] &, 96] (* _Michael De Vlieger_, Oct 05 2020 *)
%o A286385 (PARI)
%o A286385 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
%o A286385 A286385(n) = (A003961(n) - sigma(n));
%o A286385 for(n=1, 16384, write("b286385.txt", n, " ", A286385(n)));
%o A286385 (Scheme)
%o A286385 (define (A286385 n) (- (A003961 n) (A000203 n)))
%o A286385 (Python)
%o A286385 from sympy import factorint, nextprime, divisor_sigma as D
%o A286385 from operator import mul
%o A286385 def a048673(n):
%o A286385     f = factorint(n)
%o A286385     return 1 if n==1 else (1 + reduce(mul, [nextprime(i)**f[i] for i in f]))/2
%o A286385 def a(n): return 2*a048673(n) - D(n) - 1 # _Indranil Ghosh_, May 12 2017
%Y A286385 Cf. A000203, A001359, A003961, A001065, A031924, A033879, A048673, A151800, A285705, A326042, A336702, A336851, A336852, A337549 (Möbius transform).
%Y A286385 Cf. A326057 [= gcd(a(n), A252748(n))].
%K A286385 nonn
%O A286385 1,4
%A A286385 _Antti Karttunen_, May 09 2017