A395584 Least k >= 3 such that psi(k)/2 has exactly n divisors, where psi = A002322 is the reduced totient function.

3, 5, 19, 13, 128, 37, 512, 61, 73, 97, 8192, 181, 32768, 641, 1843, 241, 524288, 577, 2097152, 673, 1153, 57344, 33554432, 1009, 2593, 40961, 1801, 2689, 2147483648, 2017, 8589934592, 2161, 18433, 3670016, 10369, 2521, 549755813888, 5767169, 622592, 3361, 8796093022208

Offset

1,1

Comments

For every n >= 2, psi(2^(n+2)/2) = 2^(n-1) has exactly n divisors, so a(n) <= 2^(n+2).

For prime p, psi(a(p)) must be either 2^p or 2*P^(p-1) for some odd prime P. Since psi(p1^e1 * p2^e2 * ... * pr^er) = lcm(psi(p1^e1), ..., psi(pr^er)), a(p) must actually be a prime power, that is, either 2^(p+2), a prime of the form 2^p + 1 (only possible for p = 2), a prime of the form 2*P^(p-1) + 1, or 3^p. We see that a(p) = 2^(p+2) for p > 3.

Links

Jianing Song, Table of n, a(n) for n = 1..241

Jianing Song, A395584 on semiprimes <= 10^4.

Jianing Song, Notes on A395584.

Jianing Song, PARI Program.

Mathematica

a[n_]:=Module[{k=3}, While[DivisorSigma[0, CarmichaelLambda[k]/2]!=n, k++]; k]; Array[a, 22] (* James C. McMahon, Jun 02 2026 *)

Prog

(PARI) \\ See Links.

Crossrefs

Cf. A002322, A066800 (number of divisors of psi), A276044 (for EulerPhi), A396489 (for psi).

Sequence in context: A272019 A270723 A242961 * A025046 A305700 A299073

Adjacent sequences: A395581 A395582 A395583 * A395585 A395586 A395587

Keyword

nonn

Author

Jianing Song, May 28 2026

Status

approved