A393179 Number of k <= 2^(n-1) such that tau(k) = n where tau = A000005.

1, 1, 1, 2, 1, 5, 1, 16, 5, 13, 1, 211, 1, 35, 19, 3134, 1, 1577, 1, 8043, 46, 319, 1, 615620, 19, 1045, 1565, 383778, 1, 768107, 1, 167262047, 374, 12296, 83, 325122420, 1, 43460, 1167, 6200272135, 1, 409822597, 1, 1108940842, 352281, 564349, 1, 7832178297534

Offset

1,4

Comments

For prime p, A000005(p^(n-1)) = n.

a(n) = 1 if n is noncomposite (A008578) and a(n) > 1 if n is composite (A002808).

Links

Daniel Suteu, Table of n, a(n) for n = 1..63

Daniel Suteu, Perl program

Example

 n \   k                                                                         | a(n)
---------------------------------------------------------------------------------------
 1  |  1                                                                         |  1
 2  |  2                                                                         |  1
 3  |  4                                                                         |  1
 4  |  6, 8                                                                      |  2
 5  |  16                                                                        |  1
 6  |  12, 18, 20, 28, 32                                                        |  5
 7  |  64                                                                        |  1
 8  |  24, 30, 40, 42, 54, 56, 66, 70, 78, 88, 102, 104, 105, 110, 114, 128      | 16
 9  |  36, 100, 196, 225, 256                                                    |  5
 10 |  48, 80, 112, 162, 176, 208, 272, 304, 368, 405, 464, 496, 512             | 13

Mathematica

d[k_] := d[k] = DivisorSigma[0, k]; a[n_] := Count[Range[2^(n - 1)], _?(d[#] == n &)]; Array[a, 20] (* Amiram Eldar, Mar 11 2026 *)

Prog

(Magma) [#[k: k in [1..2^(n - 1)] | #Divisors(k) eq n]: n in [1..23]];
(PARI) a(n) = #select(x->(numdiv(x)==n), [1..2^(n-1)]); \\ Michel Marcus, Mar 11 2026
(PARI) list(nmax) = {my(v = vector(nmax), n); for(k = 1, 2^(nmax-1), n = numdiv(k); if(n <= nmax && k <= (1 << (n-1)), v[n]++)); v; } \\ Amiram Eldar, Mar 11 2026
(Perl) # See Links section

Crossrefs

Cf. A000005, A007053, A060967.

Sequence in context: A327249 A173108 A173111 * A363739 A257459 A372494

Adjacent sequences: A393176 A393177 A393178 * A393180 A393181 A393182

Keyword

nonn

Author

Juri-Stepan Gerasimov, Mar 11 2026

Extensions

a(24)-a(26) from Michel Marcus, Mar 11 2026

a(27)-a(34) from Amiram Eldar, Mar 11 2026

a(35)-a(47) from Jinyuan Wang, Apr 05 2026

a(48) from Daniel Suteu, May 18 2026

Status

approved