A349360 Number of positive integer pairs (s,t), with s,t <= n and s <= t such that either both s and t divide n or both do not.

1, 3, 4, 7, 9, 13, 18, 20, 27, 31, 48, 42, 69, 65, 76, 81, 123, 99, 156, 126, 163, 181, 234, 172, 259, 263, 286, 274, 381, 289, 438, 372, 445, 475, 506, 423, 633, 605, 640, 564, 783, 631, 864, 762, 801, 913, 1038, 796, 1087, 1011, 1138, 1102, 1329, 1117, 1336, 1212, 1441

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = A184389(n) + A335567(n). - Alois P. Heinz, Nov 15 2021

a(n) = A000005(n)*(A000005(n)-n) + n(n+1)/2. - Chai Wah Wu, Nov 19 2021

a(p) = (p^2 - 3*p + 8)/2 for primes p. - Wesley Ivan Hurt, Nov 28 2021

Example

a(5) = 9; There are 9 positive integer pairs (s,t), with s <= t such that both s and t divide 5 or both do not. They are (1,1), (1,5), (2,2), (2,3), (2,4), (3,3), (3,4), (4,4), (5,5).

Maple

a:= n-> add(add(`if`(irem(n, j)>0 xor irem(n, i)=0, 1, 0), i=1..j), j=1..n):
seq(a(n), n=1..57);  # Alois P. Heinz, Nov 15 2021

Mathematica

a[n_] := Module[{d = DivisorSigma[0, n]}, n*(n+1)/2 - d*(n-d)]; Array[a, 100] (* Amiram Eldar, Feb 04 2025 *)

Prog

(Python)
from sympy import divisor_count
def A349360(n):
    m = divisor_count(n)
    return m*(m-n) + n*(n+1)//2 # Chai Wah Wu, Nov 19 2021
(PARI) a(n) = {my(d = numdiv(n)); n*(n+1)/2 - d*(n-d); } \\ Amiram Eldar, Feb 04 2025

Crossrefs

Cf. A000005, A184389, A335567.

Sequence in context: A379157 A330146 A295069 * A103054 A140208 A098390

Adjacent sequences: A349357 A349358 A349359 * A349361 A349362 A349363

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Nov 15 2021

Status

approved