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

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, 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, 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

Offset: 1

Author

Antti Karttunen, May 09 2017

Keywords

• nonn

Comments

Are all terms nonnegative? This question is equivalent to the question posed in A285705.

From Antti Karttunen, Aug 05 2020: (Start)

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.

See comments in A326042.

(End)

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

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16383

Crossrefs

Cf. A000203, A001359, A003961, A001065, A031924, A033879, A048673, A151800, A285705, A326042, A336702, A336851, A336852, A337549 (Möbius transform).

Cf. A326057 [= gcd(a(n), A252748(n))].

Programs

Mathematica

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 *)

PARI

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A286385(n) = (A003961(n) - sigma(n));
for(n=1, 16384, write("b286385.txt", n, " ", A286385(n)));

Python

from sympy import factorint, nextprime, divisor_sigma as D
from operator import mul
def a048673(n):
    f = factorint(n)
    return 1 if n==1 else (1 + reduce(mul, [nextprime(i)**f[i] for i in f]))/2
def a(n): return 2*a048673(n) - D(n) - 1 # Indranil Ghosh, May 12 2017

Scheme

(define (A286385 n) (- (A003961 n) (A000203 n)))

Formula

a(n) = A285705(A048673(n)) - 1 = 2*A048673(n) - A000203(n) - 1.

a(n) = A336852(n) - A336851(n). - Antti Karttunen, Aug 05 2020

a(n) = Sum_{d|n} A337549(d). - Antti Karttunen, Sep 22 2020

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