A036879 If n = (p_1)^(m_1)...(p_k)^(m_k) then a(n) = (p_1)^((p_1)^(m_1) - 1)...(p_k)((p_k)^(m_k) - 1).

1, 2, 9, 8, 625, 18, 117649, 128, 6561, 1250, 25937424601, 72, 23298085122481, 235298, 5625, 32768, 48661191875666868481, 13122, 104127350297911241532841, 5000, 1058841, 51874849202, 907846434775996175406740561329, 1152

Offset

1,2

Comments

These integers are refactorable: the number of divisors divides the number itself.

Links

Amiram Eldar, Table of n, a(n) for n = 1..388

Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2 (1999), Article 99.1.2.

Simon Colton, HR - Automatic Theory Formation in Pure Mathematics. [Wayback Machine copy]

Formula

(p_1)^(m_1)...(p_k)^(m_k) -> (p_1)^((p_1)^(m_1) - 1)...(p_k)((p_k)^(m_k) - 1).

Example

a(6) = 18 because 6 = 2^(1)3^(1) -> 2^(2^(1) - 1)3^(3^(1) - 1) = 18.

Mathematica

f[p_, e_] := p^(p^e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 24] (* Amiram Eldar, Sep 24 2023 *)

Prog

(PARI) a(n) = my(f = factor(n)); for (i=1, #f~, f[i, 2] = f[i, 1]^f[i, 2] - 1); factorback(f); \\ Michel Marcus, Dec 08 2014

Crossrefs

Cf. A033950.

Sequence in context: A230283 A121067 A073904 * A281389 A073927 A198358

Adjacent sequences: A036876 A036877 A036878 * A036880 A036881 A036882

Keyword

nonn,mult

Author

Simon Colton (simonco(AT)cs.york.ac.uk)

Status

approved