A366144 a(n) = n/d(n) if d(n) | n, otherwise a(n) = n*d(n), where d(n) = A000005(n) is the number of divisors of n.

1, 1, 6, 12, 10, 24, 14, 2, 3, 40, 22, 2, 26, 56, 60, 80, 34, 3, 38, 120, 84, 88, 46, 3, 75, 104, 108, 168, 58, 240, 62, 192, 132, 136, 140, 4, 74, 152, 156, 5, 82, 336, 86, 264, 270, 184, 94, 480, 147, 300, 204, 312, 106, 432, 220, 7, 228, 232, 118, 5, 122, 248, 378, 448, 260, 528, 134

Offset: 1

Author

Neal Gersh Tolunsky, Sep 30 2023

Keywords

• nonn
• easy

Examples

			n=3 has d(3) = 2 divisors (like all primes) and 3 is not divisible by 2, so we multiply: a(3) = 3*2 = 6.
n=8 has d(8) = 4 divisors and 8 is divisible by 4, so we divide: a(8) = 8/4 = 2.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, showing primes in red, composite prime powers in gold, even squarefree semiprimes in light green, other squarefree composites in dark green, and numbers neither squarefree nor prime powers in blue. Powerful numbers that are not prime powers are highlighted in light blue.
• Neal Gersh Tolunsky, Graph of first 150000 terms.

Crossrefs

Cf. A000005, A038040, A033950, A036762.

Cf. A366067 (iterate starting 578).

Programs

Mathematica

a[n_] := n * If[Divisible[n, d = DivisorSigma[0, n]], 1/d, d]; Array[a, 100] (* Amiram Eldar, Oct 01 2023 *)

PARI

a(n) = my(d=numdiv(n)); if (n % d, n*d, n/d); \\ Michel Marcus, Oct 01 2023

Python

from sympy import divisor_count
def A366144(n): return n*d if (q:=divmod(n,d:=int(divisor_count(n))))[1] else q[0] # Chai Wah Wu, Oct 02 2023

Formula

sqrt(n)/2 <= a(n) <= 2*n*sqrt(n). - Yifan Xie, Oct 01 2023