A330492 a(n) = sum of second differences of the sorted divisors of n.

0, 0, 1, 0, 2, 0, 3, 4, 4, 0, 5, 0, 6, 8, 7, 0, 8, 0, 9, 12, 10, 0, 11, 16, 12, 16, 13, 0, 14, 0, 15, 20, 16, 24, 17, 0, 18, 24, 19, 0, 20, 0, 21, 28, 22, 0, 23, 36, 24, 32, 25, 0, 26, 40, 27, 36, 28, 0, 29, 0, 30, 40, 31, 48, 32, 0, 33, 44, 34, 0, 35, 0, 36

Offset

2,5

Comments

The sums of the first differences of the divisors of n are given by the sequence b(n) = n - 1.

Let the set {D(i)} = {d(i + 1) - d(i)} where the d(i) are the divisors of an integer m listed in ascending order with i = 1, 2 , ..., tau(n)-1. The sequence is given by a(n) = Sum_{k = 1..tau(n)-2} (D(k + 1) - D(k)).

Links

Antti Karttunen, Table of n, a(n) for n = 2..8191

Antti Karttunen, Data supplement: n, a(n) computed for n = 2..65537

Formula

a(n) = d(tau(n)) - d(tau(n) - 1) + d(1) - d(2) where d(i) are the divisors of n.

a(prime(n)) = 0 and a(2k) = k-1, k = 1, 2, ...

a(p^2) = (p-1)^2 if p prime, with the generalization a(p^m) = (p-1)(p^(m-1) - 1).

a(n) = (n/p-1)*(p-1), where p is the least prime factor of n. - Nathaniel Gregg, Apr 04 2021

Example

a(12) = 5 because the divisors of 12 are {1, 2, 3, 4, 6, 12} and {D(i)} = {d(i+1)-d(i)} ={1, 1, 1, 2, 6}, Sum_{D(i), i = 1..4} {D(i+1)-D(i)} = 0 + 0 + 1 + 4 = 5.

Maple

with(numtheory):nn:=100:
for n from 2 to nn do:
d:=divisors(n):n0:=nops(d):T:=array(1..n0-1, [0$n0-1]):
  for j from 1 to n0-1 do:
   T[j]:=d[j+1]-d[j]:
  od:
   s:=sum(‘T[i+1]-T[i] ’, ‘i’=1..n0-2): printf(`%d, `, s):
od:
*** alternative program using the formula ***
with(numtheory):nn:=100:
for n from 2 to nn do:
d:=divisors(n):t:=tau(n):s:=d[t]-d[t-1]+d[1]-d[2] :
  printf(`%d, `, s):
od:

Mathematica

Array[Total@ Differences[Divisors@ #, 2] &, 73, 2] (* Michael De Vlieger, Dec 16 2019 *)

Prog

(PARI) a(n) = my(d=divisors(n)); d[#d] - d[#d-1] + d[1] - d[2]; \\ Michel Marcus, Feb 05 2020
(Python)
from sympy import primefactors
def a(n): p = primefactors(n)[0]; return (n//p - 1) * (p - 1)
print([a(n) for n in range(2, 75)]) # Michael S. Branicky, Apr 04 2021

Crossrefs

Cf. A000005, A027750, A036263, A193829.

Sequence in context: A285722 A274441 A213859 * A350534 A101336 A300001

Adjacent sequences: A330489 A330490 A330491 * A330493 A330494 A330495

Keyword

nonn

Author

Michel Lagneau, Dec 16 2019

Status

approved