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A127492 Indices m of primes such that Sum_{k=0..2, k prime(m+k)*prime(m+j)*prime(m+j+1) is twice a prime.

%I A127492 #27 Aug 02 2023 14:37:29
%S A127492 2,10,17,49,71,72,75,145,161,167,170,184,244,250,257,266,267,282,286,
%T A127492 301,307,325,343,391,405,429,450,537,556,561,584,685,710,743,790,835,
%U A127492 861,904,928,953
%N A127492 Indices m of primes such that Sum_{k=0..2, k<j<=3} prime(m+k)*prime(m+j)*prime(m+j+1) is twice a prime.
%C A127492 Let p_0 .. p_4 be five consecutive primes, starting with the m-th prime. The index m is in the sequence if the absolute value [x^0] of the polynomial (x-p_0)*[(x-p_1)*(x-p_2) + (x-p_2)*(x-p_3) + (x-p_3)*(x-p_4)] + (x-p_1)*[(x-p_2)*(x-p_3) + (x-p_3)*(x-p_4)] + (x-p_2)*(x-p_3)*(x-p_4) is two times a prime. The correspondence with A127491: the coefficient [x^2] of the polynomial (x-p_0)*(x-p_1)*..*(x-p_4) is the sum of 10 products of a set of 3 out of the 5 primes. Here the sum is restricted to the 6 products where the two largest of the 3 primes are consecutive. - _R. J. Mathar_, Apr 23 2023
%H A127492 Harvey P. Dale, <a href="/A127492/b127492.txt">Table of n, a(n) for n = 2..1000</a>
%p A127492 isA127492 := proc(k)
%p A127492     local x,j ;
%p A127492     (x-ithprime(k))* mul( x-ithprime(k+j),j=1..2)
%p A127492     +(x-ithprime(k))* mul( x-ithprime(k+j),j=2..3)
%p A127492     +(x-ithprime(k))* mul( x-ithprime(k+j),j=3..4)
%p A127492     +(x-ithprime(k+1))* mul( x-ithprime(k+j),j=2..3)
%p A127492     +(x-ithprime(k+1))* mul( x-ithprime(k+j),j=3..4)
%p A127492     +(x-ithprime(k+2))* mul( x-ithprime(k+j),j=3..4) ;
%p A127492     p := abs(coeff(expand(%/2),x,0)) ;
%p A127492     if type(p,'integer') then
%p A127492         isprime(p) ;
%p A127492     else
%p A127492         false ;
%p A127492     end if ;
%p A127492 end proc:
%p A127492 for k from 1 to 900 do
%p A127492     if isA127492(k) then
%p A127492         printf("%a,",k) ;
%p A127492     end if ;
%p A127492 end do: # _R. J. Mathar_, Apr 23 2023
%t A127492 a = {}; Do[If[PrimeQ[(Prime[x] Prime[x + 1]Prime[x + 2] + Prime[x] Prime[x + 2]Prime[x + 3] + Prime[x] Prime[x + 3] Prime[x + 4] + Prime[x + 1] Prime[x + 2]Prime[x + 3] + Prime[x + 1] Prime[x + 3]Prime[x + 4] + Prime[x + 2] Prime[x + 3] Prime[x + 4])/2], AppendTo[a, x]], {x, 1, 1000}]; a
%t A127492 prQ[{a_,b_,c_,d_,e_}]:=PrimeQ[(a b c+a c d+a d e+b c d+b d e+c d e)/2]; PrimePi/@Select[ Partition[ Prime[Range[1000]],5,1],prQ][[;;,1]] (* _Harvey P. Dale_, Apr 21 2023 *)
%Y A127492 Cf. A001043, A034961, A034963, A034964, A127333, A127334, A127335, A127336, A127337, A127338, A127339, A127340, A127341, A127342, A127343, A127345, A127346, A127347, A127348, A127349, A127351, A037171, A034962, A034965, A082246, A082251, A070934, A006094, A046301, A046302, A046303, A046324, A046325, A046326, A046327, A127489, A127490, A127491.
%K A127492 nonn,uned,obsc
%O A127492 1,1
%A A127492 _Artur Jasinski_, Jan 16 2007
%E A127492 Definition simplified by _R. J. Mathar_, Apr 23 2023
%E A127492 Edited by _Jon E. Schoenfield_, Jul 23 2023