A367326 -id:A367326 - cp's OEIS frontend

A367327 a(n) = Sum_{(n - k) does not divide n, 0 <= k < n} k^2.

0, 0, 0, 1, 1, 14, 5, 55, 39, 104, 115, 285, 104, 506, 457, 575, 611, 1240, 790, 1785, 1204, 1950, 2349, 3311, 1746, 3924, 4155, 4625, 4312, 6930, 4375, 8555, 6939, 9032, 10127, 10845, 7887, 14910, 14549, 15603, 12730, 20540, 15273, 23821, 20648, 21874, 26905

Offset: 0

Peter Luschny, Nov 14 2023

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A367327, A000330, A024816.

divides := (k, n) -> k = n or (k > 0 and irem(n, k) = 0):
A367327 := n -> local k; add(`if`(divides(n - k, n), 0, k^2), k = 0..n - 1):
seq(A367327(n), n = 0..61);

a[n_] := Sum[If[Divisible[n, n - k], 0, k^2], {k, 0, n - 1}]
Table[a[n], {n, 0, 46}]

a(n) = sum(k=0, n-1, if (n % (n-k), k^2)); \\ Michel Marcus, Nov 15 2023

from math import prod
from sympy import factorint
def A367327(n):
    f = factorint(n).items()
    return n*(n-1)*((n<<1)-1)//6-n*(n*prod(e+1 for p,e in f)-(prod((p**(e+1)-1)//(p-1) for p,e in f)<<1))-prod((p**(e+1<<1)-1)//(p**2-1) for p,e in f) if n else 0 # Chai Wah Wu, Nov 14 2023

def A367327(n): return sum(k^2 for k in (0..n-1) if not (n-k).divides(n))
print([A367327(n) for n in range(47)])

A367326(n+1) + a(n+1) = A000330(n).

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

Original entry on oeis.org

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

Peter Luschny, Nov 15 2023

nonn

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.

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

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

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

# 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);

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

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)])

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

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)])

An additive decomposition of the triangular numbers:

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

A367493 a(n) = Sum_{d|n} (n-d)^n.

Original entry on oeis.org

0, 1, 8, 97, 1024, 20450, 279936, 7509953, 144295424, 4570291850, 100000000000, 4491754172274, 106993205379072, 5221973073321002, 171975117132398592, 8931527427394008833, 295147905179352825856, 20290116242888952838355, 708235345355337676357632, 51879761166564630630389778

Offset: 1

Chai Wah Wu, Nov 20 2023

nonn

Cf. A094471, A367326, A367327, A367368.

a[n_]:=Sum[(n-d)^n,{d,Divisors[n]}]; Array[a,20] (* Stefano Spezia, Nov 20 2023 *)

from sympy import divisors
def A367493(n): return sum((n-d)**n for d in divisors(n, generator=True))

a(n) = Sum_{k=0..n} (-1)^k*binomial(n,k)*n^(n-k)*sigma_k(n).

cp's OEIS Frontend

A367327 a(n) = Sum_{(n - k) does not divide n, 0 <= k < n} k^2.

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

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A367493 a(n) = Sum_{d|n} (n-d)^n.

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