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A245589 Primes which are the average of the two adjacent primes and also of the two adjacent squarefree numbers.

%I A245589 #25 Nov 10 2024 05:16:13
%S A245589 53,593,1747,2287,4013,4409,5563,6317,8117,10657,10853,11933,12547,
%T A245589 12583,12653,15161,16937,17047,17851,18341,19603,19949,20107,22051,
%U A245589 26693,31051,32993,35851,35911,39113,42209,42533,44041,46889,47527,48259,50417,51461
%N A245589 Primes which are the average of the two adjacent primes and also of the two adjacent squarefree numbers.
%C A245589 Intersection of A006562 and A240475. Intersection of A006562 and A245289.
%H A245589 Jens Kruse Andersen, <a href="/A245589/b245589.txt">Table of n, a(n) for n = 1..10000</a>
%e A245589 53 is in this sequence because 53 = prime(16) = (prime(15) + prime(17))/2 = (47 + 59)/2 and 53 = squarefree(33) = (squarefree(32) + squarefree(34))/2 = (51 + 55)/2.
%p A245589 Primes:= select(isprime,[$1..10^5]):
%p A245589 Sqfree:= select(numtheory:-issqrfree,[$1..10^5]):
%p A245589 A:= NULL:
%p A245589 for i from 2 to nops(Primes)-1 do
%p A245589    if Primes[i] = (Primes[i+1]+Primes[i-1])/2 then
%p A245589       member(Primes[i],Sqfree,'j');
%p A245589       if Primes[i] = (Sqfree[j-1]+Sqfree[j+1])/2 then
%p A245589          A:= A,Primes[i]
%p A245589       fi
%p A245589    fi
%p A245589 od:
%p A245589 A; # _Robert Israel_, Aug 21 2014
%o A245589 (PARI)
%o A245589 maxp=60000;
%o A245589 p=[]; my(v=primes(maxp)); for(k=2, #v-1, if(2*v[k] == v[k-1]+v[k+1], p=concat(p, v[k]))); p;
%o A245589 v = select(n->issquarefree(n), vector(maxp, n, n));
%o A245589 s=[]; for(k=2, #v-1, if(2*v[k] == v[k-1]+v[k+1], s=concat(s, v[k]))); s;
%o A245589 setintersect(p, s) \\ _Colin Barker_, Aug 07 2014
%Y A245589 Cf. A006562, A240475, A245289.
%K A245589 nonn
%O A245589 1,1
%A A245589 _Juri-Stepan Gerasimov_, Jul 26 2014
%E A245589 Missing term (16937) inserted by _Colin Barker_, Aug 07 2014