A359062 Nonprime terms of A359059.
1, 8, 9, 18, 20, 27, 32, 36, 42, 44, 45, 49, 50, 54, 63, 68, 72, 78, 80, 81, 84, 90, 92, 99, 105, 108, 110, 114, 116, 117, 125, 126, 128, 135, 144, 153, 156, 162, 164, 168, 169, 170, 171, 176, 180, 186, 188, 189, 195, 198, 200, 207, 210, 212, 216, 222, 225, 228, 230
Offset: 1
Keywords
Examples
8 is a term because 3|(4+2+12). 9 is a term because 3|(6+3+12). 18 is a term because 3|(6+8+36). 20 is a term because 3|(8+10+36). 27 is a term because 3|(18+3+36).
Programs
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Maple
filter:= proc(n) local F,p,ph,r,ps; F:= numtheory:-factorset(n); if F = {n} then return false fi; ph:= n * mul((p-1)/p, p = F); r:= convert(F,`*`); ps:= n * mul((p+1)/p, p = F); (ph+r+ps) mod 3 = 0 end proc: select(filter, [$1..1000]); # Robert Israel, Dec 20 2022 -
Mathematica
q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Divisible[Times @@ ((p - 1)*p^(e - 1)) + Times @@ p + Times @@ ((p + 1)*p^(e - 1)), 3]]; Select[Range[230], ! PrimeQ[#] && q[#] &] (* Amiram Eldar, Dec 20 2022 *) -
PARI
isok(m) = !isprime(m) && (((eulerphi(m) + factorback(factorint(m)[, 1]) + m*sumdiv(m, d, moebius(d)^2/d)) % 3) == 0); \\ Michel Marcus, Dec 27 2022
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Python
from sympy.ntheory.factor_ import totient from sympy import isprime, primefactors, prod def rad(n): return 1 if n < 2 else prod(primefactors(n)) def psi(n): plist = primefactors(n) return n*prod(p+1 for p in plist)//prod(plist) # Output display terms. for n in range(1,231): if(False == isprime(n)): if(0 == (totient(n) + rad(n) + psi(n)) % 3): print(n, end = ", ")
Comments