A367175 a(n) = Sum_{d|n} (-1)^[d is prime] * d, where [] denotes the Iverson bracket.

1, -1, -2, 3, -4, 2, -6, 11, 7, 4, -10, 18, -12, 6, 8, 27, -16, 29, -18, 28, 12, 10, -22, 50, 21, 12, 34, 38, -28, 52, -30, 59, 20, 16, 24, 81, -36, 18, 24, 76, -40, 72, -42, 58, 62, 22, -46, 114, 43, 79, 32, 68, -52, 110, 40, 102, 36, 28, -58, 148, -60, 30

Offset

1,3

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Formula

{k: a(k) < 0} = {A000040}.

{k: a(k) > k} = {A033942}.

{k: a(k) < k} = {A037143} \ {1}.

sigma(n) - a(n) = 2 * A008472(n).

Conjecture: {k: a(k) divides sigma(k)} = {1, 2, 3, 6, 14, 15, 35}.

Maple

Isprime := n -> if isprime(n) then 1 else 0 fi:
a := n -> local d; add((-1)^Isprime(d) * d, d in NumberTheory:-Divisors(n)):
seq(a(n), n = 1..62);

Mathematica

Array[DivisorSum[#, (-1)^Boole[PrimeQ[#]]*# &] &, 62] (* Michael De Vlieger, Nov 10 2023 *)

Prog

(SageMath)
def A367175(n): return sum((-1)^is_prime(d)*d for d in divisors(n))
print([A367175(n) for n in range(1, 63)])
(PARI) a(n) = sumdiv(n, d, (-1)^isprime(d)*d); \\ Michel Marcus, Nov 10 2023
(Python)
from sympy import divisor_sigma, primefactors
def A367175(n): return divisor_sigma(n)-(sum(primefactors(n))<<1) # Chai Wah Wu, Nov 10 2023

Crossrefs

Cf. A000203, A000040, A033942, A037143, A008472, A367176.

Sequence in context: A384053 A384051 A340368 * A153038 A368698 A324911

Adjacent sequences: A367172 A367173 A367174 * A367176 A367177 A367178

Keyword

sign

Author

Peter Luschny, Nov 08 2023

Status

approved