A367368 a(n) = Sum_{(n - k) does not divide n, 0 <= k <= n} k.

0, 1, 2, 4, 5, 11, 9, 22, 19, 31, 33, 56, 34, 79, 73, 84, 87, 137, 102, 172, 132, 179, 201, 254, 168, 281, 289, 310, 294, 407, 297, 466, 399, 477, 513, 538, 433, 667, 649, 680, 590, 821, 663, 904, 810, 843, 969, 1082, 820, 1135, 1068, 1194, 1164, 1379, 1173

Offset: 0

Author

Peter Luschny, Nov 15 2023

Keywords

• nonn

Comments

The case n = 0 is well defined because zero divides zero. When implementing the sequence it is advisable to use the definition of divisibility of an integer directly and not the set of divisors, because this is infinite in the case n = 0 and, therefore, cannot be represented in computer algebra systems, which leads to a wide variety of error messages depending on the system. Some of these error messages are in turn incorrect, because the test of divisibility by zero does not involve division and therefore should not lead to a 'ZeroDivisionError' or similar.

References

• Tom M. Apostol, Introduction to Analytic Number Theory, Springer 1976, p. 14.

Crossrefs

Cf. A094471 (the not negated case), A000217.

Programs

Julia

using AbstractAlgebra
function A367326(n) sum(k for k in 0:n if ! is_divisible_by(n, n - k)) end
[A367326(n) for n in 0:54] |> println

Maple

# Warning: Be careful when using the deprecated 'numtheory' package.
# It might not handle the case n = 0 correctly. A better solution is:
divides := (k, n) -> k = n or (k > 0 and irem(n, k) = 0):
A367368 := n -> local k; add(`if`(divides(n - k, n), 0, k), k = 0..n):
seq(A367368(n), n = 0..61);

Mathematica

a[n_]:=n+Sum[k*Boole[!Divisible[n,n-k]],{k,0,n-1}]; Array[a,55,0] (* Stefano Spezia, Nov 15 2023 *)

Python

def divides(k, n): return k == n or ((k > 0) and (n % k == 0))
def A367368(n): return sum(k for k in range(n + 1) if not divides(n - k, n))
print([A367368(n) for n in range(55)])

Python

from math import prod
from sympy import factorint
def A367368(n):
    f = factorint(n).items()
    return (n*(n+1)>>1)-n*prod(e+1 for p,e in f)+prod((p**(e+1)-1)//(p-1) for p,e in f) if n else 0 # Chai Wah Wu, Nov 17 2023

SageMath

def A367368(n): return sum(k for k in (0..n) if not (n - k).divides(n))
print([A367368(n) for n in range(55)])

Formula

An additive decomposition of the triangular numbers:

a(n) + A094471(n) = A000217(n) for n >= 0 assuming A094471 with correct offset 0.