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%I #27 Aug 02 2023 14:37:29
%S 2,10,17,49,71,72,75,145,161,167,170,184,244,250,257,266,267,282,286,
%T 301,307,325,343,391,405,429,450,537,556,561,584,685,710,743,790,835,
%U 861,904,928,953
%N 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 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 Harvey P. Dale, <a href="/A127492/b127492.txt">Table of n, a(n) for n = 2..1000</a>
%p isA127492 := proc(k)
%p local x,j ;
%p (x-ithprime(k))* mul( x-ithprime(k+j),j=1..2)
%p +(x-ithprime(k))* mul( x-ithprime(k+j),j=2..3)
%p +(x-ithprime(k))* mul( x-ithprime(k+j),j=3..4)
%p +(x-ithprime(k+1))* mul( x-ithprime(k+j),j=2..3)
%p +(x-ithprime(k+1))* mul( x-ithprime(k+j),j=3..4)
%p +(x-ithprime(k+2))* mul( x-ithprime(k+j),j=3..4) ;
%p p := abs(coeff(expand(%/2),x,0)) ;
%p if type(p,'integer') then
%p isprime(p) ;
%p else
%p false ;
%p end if ;
%p end proc:
%p for k from 1 to 900 do
%p if isA127492(k) then
%p printf("%a,",k) ;
%p end if ;
%p end do: # _R. J. Mathar_, Apr 23 2023
%t 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 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 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 nonn,uned,obsc
%O 1,1
%A _Artur Jasinski_, Jan 16 2007
%E Definition simplified by _R. J. Mathar_, Apr 23 2023
%E Edited by _Jon E. Schoenfield_, Jul 23 2023
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