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A336486 The least prime s for which there exist primes p, q, r such that phi(p*q*s^n) = phi(r*s^(2n+1)) and sigma(p*q*s^n) = sigma(r*s^(2n+1)).

%I A336486 #20 Sep 30 2020 09:05:04
%S A336486 2,2,2,2,5,37,13,7,2,19,7,7,2,19,4447,2,2,2,2,5,73,23,37,2,2,19,19,2,2
%N A336486 The least prime s for which there exist primes p, q, r such that phi(p*q*s^n) = phi(r*s^(2n+1)) and sigma(p*q*s^n) = sigma(r*s^(2n+1)).
%C A336486 This sequence is an inversion of A336485.
%C A336486 Let "a and b are similar" mean that for positive integers a and b we have phi(a) = phi(b), tau(a) = tau(b) and sigma(a) = sigma(b).
%C A336486 Conjecture: For each positive integer n there are infinitely many primes s such that numbers p*q*s^n and r*s^(2n+1) are similar for some primes p, q, r.
%e A336486 a(2) = 2 because for n = 2 and prime s = 2, a = 19*89*s^n and b = 199*s^(2n+1), we have phi(a) = phi(b) and sigma(a) = sigma(b).
%p A336486 with(NumberTheory):
%p A336486 KS := []; for k to 29 do tf := false;
%p A336486 for ii do s := ithprime(ii); c := 2*s^(k+1)+1; cc := (c^2-1)*(1/2); Q := Divisors(cc);
%p A336486 for d in Q do q := d+c; if isprime(q) then p := c+cc/(q-c); if p < q then break end if;
%p A336486 if isprime(p) then r := 2*(p+q)-c; if isprime(r) then print([k, [p, q], r], s); KS := [op(KS), s]; tf := true; break end if end if end if end do;
%p A336486 if tf then break end if end do end do; KS
%o A336486 (PARI) is(t, u, x, y) = ispseudoprime(t*x+1) && ispseudoprime(u*y/t+1) && ispseudoprime(x*y+1);
%o A336486 a(n) = {my(s=1, t, u); while(s=nextprime(s+1), for(i=0, 1+n\2, t=s^i; fordiv(2*(1+u=s^(n+1)), d, if(is(t, u, 2*u/t+d, 2*t+(2*u+2)/d) || is(t, u, 2*u/t-d, 2*t-(2*u+2)/d), return(s))))); } \\ _Jinyuan Wang_, Sep 30 2020
%Y A336486 Cf. A000005, A000010, A000203, A134922, A336485.
%K A336486 nonn,more
%O A336486 1,1
%A A336486 _Vladimir Letsko_, Jul 23 2020