cp's OEIS Frontend

A236255 Prime numbers s for which there exist primes p, q, r such that phi(pqs)=phi(rs^3), sigma(pqs)=sigma(rs^3).

%I A236255 #10 Feb 05 2014 18:11:51
%S A236255 2,3,5,7,31,43,139,157,191,269,293,337,463,557,593,683,709,757,769,
%T A236255 983,1021,1567,1583,2293,2309,2689,2707,2801,2917,3319,3323,3583,3823,
%U A236255 4271,5507,5557,6037,6043,6079,6151,6469,6779,6959,6977,7207,7963,8419,8429,8521,8627,8663,8861,8887,9677,9769,10163,10613,10847,11003
%N A236255 Prime numbers s for which there exist primes p, q, r such that phi(pqs)=phi(rs^3), sigma(pqs)=sigma(rs^3).
%C A236255 Obviously tau(pqs) = tau(rs^3). So we have pairs of terms of A134922.
%C A236255 s = 593 is the least number such that there are just two matching pairs: (593*381187517*703949, 593^3*763079633) and (593*3911429*780389, 593^3*8680337). And for s = 853693 there are as many as 3 matching pairs.
%H A236255 Vladimir Letsko, <a href="http://www-old.fizmat.vspu.ru/doku.php?id=marathon:problem_163">Math Marathon, problem 163</a> (in Russian)
%e A236255 2 is in the sequence because for p = 11, q = 29, r = 71 we have phi(pqs)=phi(rs^3) and sigma(pqs)=sigma(rs^3).
%p A236255 is_A236255:=proc(s::prime) local f,Q, c,d,cc,p,q,r;
%p A236255 f:=false:c:=2*s^2+1:
%p A236255 cc:=(c^2-1)/2;
%p A236255 Q:=numtheory[divisors](cc):
%p A236255 for d in Q do q:=d+c:
%p A236255 if isprime(q) then
%p A236255 p:=c+cc/(q-c): if p<q then break fi:
%p A236255 if isprime(p) then r:=2*(p+q)-c:
%p A236255 if isprime(r) then f:=true:break fi fi
%p A236255 fi od; f end;
%p A236255 for i from 1 to 2500 do s:=ithprime(i):if is_A236255(s) then print(s) fi od:
%Y A236255 cf. A134922, A000010, A000203
%K A236255 nonn
%O A236255 1,1
%A A236255 _Vladimir Letsko_, Jan 21 2014