A039788 Numbers k such that phi(k) is equal to the product of (the sum of prime factors and the sum of exponents) of k.

9, 16, 35, 45, 150, 154, 234, 264

Offset

1,1

Comments

Next term if it exists is greater than 10^7. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 29 2004

a(9) > 10^10, if it exists. - Amiram Eldar, Jun 10 2025

Links

Table of n, a(n) for n=1..8.

Example

45 is a term since phi(45) = 24, 45 = 3^2*5^1, (3+5)*(2+1) = 24.

Mathematica

q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p-1) * p^(e-1)) == Total[p] * Total[e]]; Select[Range[2, 300], q] (* Amiram Eldar, Jun 10 2025 *)

Prog

(PARI) for(n=1, 10000000, f=factor(n); l=#f[, 1]; if(eulerphi(n)==sum(i=1, l, f[i, 1])*sum(i=1, l, f[i, 2]), print1(n, ", "))) (Ronaldo)
(PARI) isok(k) = if(k == 1, 0, my(f = factor(k)); eulerphi(f) == vecsum(f[, 1]) * vecsum(f[, 2])); \\ Amiram Eldar, Jun 10 2025

Crossrefs

Cf. A000010, A039697, A039774, A039789.

Sequence in context: A369642 A162616 A153362 * A175652 A348233 A110833

Adjacent sequences: A039785 A039786 A039787 * A039789 A039790 A039791

Keyword

nonn,more

Author

Olivier Gérard

Status

approved