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

Author

Olivier Gérard

Keywords

• nonn
• more

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

Examples

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

Crossrefs

Cf. A000010, A039697, A039774, A039789.

Programs

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 *)

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