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

1,3

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.

Formula

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

Cf. A000005, A038040, A033950, A036762.

Cf. A366067 (iterate starting 578).

Sequence in context: A315566 A236926 A134383 * A272305 A114797 A048726

Adjacent sequences: A366141 A366142 A366143 * A366145 A366146 A366147

Keyword

nonn,easy

Author

Neal Gersh Tolunsky, Sep 30 2023

Status

approved