1,1

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

Harvey P. Dale, Table of n, a(n) for n = 1..999

isA127492 := proc(k)

local x, j ;

(x-ithprime(k))* mul( x-ithprime(k+j), j=1..2)

+(x-ithprime(k))* mul( x-ithprime(k+j), j=2..3)

+(x-ithprime(k))* mul( x-ithprime(k+j), j=3..4)

+(x-ithprime(k+1))* mul( x-ithprime(k+j), j=2..3)

+(x-ithprime(k+1))* mul( x-ithprime(k+j), j=3..4)

+(x-ithprime(k+2))* mul( x-ithprime(k+j), j=3..4) ;

p := abs(coeff(expand(%/2), x, 0)) ;

if type(p, 'integer') then

isprime(p) ;

else

false ;

end if ;

end proc:

for k from 1 to 900 do

if isA127492(k) then

printf("%a, ", k) ;

end if ;

end do: # R. J. Mathar, Apr 23 2023

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

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 *)

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.

Sequence in context: A304805 A379928 A214086 * A258974 A366508 A077247

Adjacent sequences: A127489 A127490 A127491 * A127493 A127494 A127495

nonn,uned,obsc

Artur Jasinski, Jan 16 2007

Definition simplified by R. J. Mathar, Apr 23 2023

Edited by Jon E. Schoenfield, Jul 23 2023

approved