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.

2, 10, 17, 49, 71, 72, 75, 145, 161, 167, 170, 184, 244, 250, 257, 266, 267, 282, 286, 301, 307, 325, 343, 391, 405, 429, 450, 537, 556, 561, 584, 685, 710, 743, 790, 835, 861, 904, 928, 953

Offset

1,1

Comments

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

Links

Harvey P. Dale, Table of n, a(n) for n = 2..1000

Maple

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

Mathematica

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 *)

Crossrefs

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: A304809 A304805 A214086 * A258974 A366508 A077247

Adjacent sequences: A127489 A127490 A127491 * A127493 A127494 A127495

Keyword

nonn,uned,obsc

Author

Artur Jasinski, Jan 16 2007

Extensions

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

Edited by Jon E. Schoenfield, Jul 23 2023

Status

approved