A039774
Numbers k such that phi(k) is equal to the product of (the sum of prime factors and the sum of exponents) of k-1.
Original entry on oeis.org
3, 5, 9, 25, 31, 57, 116, 144, 154, 288, 372, 414, 624, 792, 10032
Offset: 1
25 is a term since phi(25) = 20, 24 = 2^3*3^1, (2+3)*(3+1) = 20.
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s[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;;, 1]]; e = f[[;;, 2]]; Total[p] * Total[e]]; Select[Range[3, 12000], EulerPhi[#] == s[#-1] &] (* Amiram Eldar, Jun 10 2025 *)
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isok(k) = if(k < 3, 0, my(f = factor(k-1)); eulerphi(k) == vecsum(f[,1]) * vecsum(f[,2])); \\ Amiram Eldar, Jun 10 2025
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.
Original entry on oeis.org
9, 16, 35, 45, 150, 154, 234, 264
Offset: 1
45 is a term since phi(45) = 24, 45 = 3^2*5^1, (3+5)*(2+1) = 24.
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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 *)
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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)
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isok(k) = if(k == 1, 0, my(f = factor(k)); eulerphi(f) == vecsum(f[,1]) * vecsum(f[,2])); \\ Amiram Eldar, Jun 10 2025
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