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A236255 Prime numbers s for which there exist primes p, q, r such that phi(pqs)=phi(rs^3), sigma(pqs)=sigma(rs^3). 1

%I

%S 2,3,5,7,31,43,139,157,191,269,293,337,463,557,593,683,709,757,769,

%T 983,1021,1567,1583,2293,2309,2689,2707,2801,2917,3319,3323,3583,3823,

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

%C Obviously tau(pqs) = tau(rs^3). So we have pairs of terms of A134922.

%C 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 Vladimir Letsko, <a href="http://www-old.fizmat.vspu.ru/doku.php?id=marathon:problem_163">Math Marathon, problem 163</a> (in Russian)

%e 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 is_A236255:=proc(s::prime) local f,Q, c,d,cc,p,q,r;

%p f:=false:c:=2*s^2+1:

%p cc:=(c^2-1)/2;

%p Q:=numtheory[divisors](cc):

%p for d in Q do q:=d+c:

%p if isprime(q) then

%p p:=c+cc/(q-c): if p<q then break fi:

%p if isprime(p) then r:=2*(p+q)-c:

%p if isprime(r) then f:=true:break fi fi

%p fi od; f end;

%p for i from 1 to 2500 do s:=ithprime(i):if is_A236255(s) then print(s) fi od:

%Y cf. A134922, A000010, A000203

%K nonn

%O 1,1

%A _Vladimir Letsko_, Jan 21 2014

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Last modified June 19 07:26 EDT 2021. Contains 345126 sequences. (Running on oeis4.)