A334628 Total area of all distinct rectangles whose length and width are relatively prime and L + W = n.

0, 1, 2, 3, 10, 5, 28, 22, 42, 30, 110, 46, 182, 91, 140, 172, 408, 159, 570, 260, 420, 385, 1012, 380, 1050, 650, 1098, 770, 2030, 620, 2480, 1368, 1760, 1496, 2380, 1290, 4218, 2109, 2964, 2120, 5740, 1806, 6622, 3190, 4020, 3795, 8648, 3064, 8428, 4150, 6800, 5356, 12402

Offset

1,3

Comments

Sum of the products of the parts in each partition of n into two relatively prime parts.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

René Gy, The sum of product pairs of integers prime to n, Math StackExchange.

Formula

a(n) = Sum_{i=1..floor(n/2)} [gcd(i,n-i) = 1] * i * (n-i), where [ ] is the Iverson bracket.

a(n) = n*A066840(n) - A295574(n). For n != 2, a(n) = n*(n*A000010(n)-A023900(n))/12 = (n*A023896(n)-A053818(n))/2. - Chai Wah Wu, Apr 28 2026

For n > 2, a(n) = (A000010(n)/12)*(n^2-A062953(n)). - Jamie Morken, May 25 2026

Mathematica

Table[Sum[i*(n - i) KroneckerDelta[GCD[i, n - i], 1], {i, Floor[n/2]}], {n, 60}]

Prog

(Python)
from math import prod
from sympy import primefactors
def A334628(n):
    ps = primefactors(n)
    return (n==2)+n*(n**2*(k:=prod(p-1 for p in ps))//prod(ps)+(k if len(ps)&1 else -k))//12 # Chai Wah Wu, Apr 28 2026

Crossrefs

Cf. A023022, A066840, A295574, A000010, A023900, A023896, A053818.

Sequence in context: A328613 A064946 A078730 * A292239 A373641 A163767

Adjacent sequences: A334625 A334626 A334627 * A334629 A334630 A334631

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Sep 09 2020

Status

approved