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%I #26 Jan 25 2023 16:04:48
%S 3,14,42,210,3570,43890,746130,14804790,281291010,8720021310,
%T 278196808890,8624101075590,353588144099190,25104758231042490,
%U 2234323482562781610,129325924468711040070,9182140637278483844970,725389110345000223752630,51501592227099266198116170
%N a(n) is the smallest k such that psi(k) and phi(k) have same distinct prime factors when k is the product of n distinct primes (psi(k) = A001615(k) and phi(k) = A000010(k)), or 0 if no such k exists.
%H Daniel Suteu, <a href="/A291138/b291138.txt">Table of n, a(n) for n = 1..36</a>
%e a(5) = 3570 = 2*3*5*7*17 because psi(3570) = 3*4*6*8*18 = 2^7*3^4, and phi(3570) = 2*4*6*16 = 2^8*3^1 and 3570 is the least number with 5 distinct prime factors having this property.
%t Rest@ Values[#][[All, 1]] &@ KeySort@ PositionIndex@ Table[If[SameQ @@ #, PrimeNu@ n, 0] &@ Map[FactorInteger[#][[All, 1]] &, {EulerPhi@ n, n Sum[MoebiusMu[d]^2/d, {d, Divisors@ n}]}], {n, 10^6}] (* _Michael De Vlieger_, Aug 26 2017, after _Michael Somos_ at A001615 *)
%o (PARI)
%o generate(A, B, n) = A=max(A, vecprod(primes(n))); (f(m, p, j, phi=1, psi=1) = my(list=List()); my(s=sqrtnint(B\m, j)); if(j==1, forprime(q=max(p, ceil(A/m)), s, if(factorback(factor((q-1)*phi)[, 1]) == factorback(factor((q+1)*psi)[, 1]), listput(list, m*q))), forprime(q=p, s, my(t=m*q); list=concat(list, f(t, q+1, j-1, phi*(q-1), psi*(q+1))))); list); vecsort(Vec(f(1, 2, n)));
%o a(n) = my(x=vecprod(primes(n)), y=2*x); while(1, my(v=generate(x, y, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ _Daniel Suteu_, Jan 25 2023
%Y Cf. A000010 (phi), A005117 (squarefree), A001615 (psi), A007947 (radical).
%K nonn
%O 1,1
%A _Altug Alkan_, Aug 18 2017
%E a(10) from _Giovanni Resta_, Aug 26 2017
%E a(11)-a(19) from _Daniel Suteu_, Jan 25 2023
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