cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A127489 a(n) is the coefficient of the linear term in the polynomial (x-prime(n))*(x-prime(n+1))*(x-prime(n+2))*(x-prime(n+3))*(x-prime(n+4)).

Original entry on oeis.org

2927, 12673, 48457, 136489, 342889, 745945, 1480489, 2760049, 5070049, 8292889, 12185065, 18656761, 27138729, 37294369, 53106049, 73698049, 95048089, 120087129, 153503149, 192747937, 247731385, 321039529, 396584569, 485290729
Offset: 1

Views

Author

Artur Jasinski, Jan 16 2007

Keywords

Comments

Arithmetic derivative (see A003415) of prime(n)*prime(n+1)*prime(n+2)*prime(n+3)*prime(n+4). [Giorgio Balzarotti, May 26 2011]

Examples

			a(1) is the coefficient of the linear term of (x-2)*(x-3)*(x-5)*(x-7)*(x-11).
This polynomial is -2310 + 2927*x - 1358*x^2 + 288*x^3 - 28*x^4 + x^5, the coefficient of the linear term equals 2927; hence a(1) = 2927.

Crossrefs

Cf. A127490.

Programs

  • Maple
    A127489 := proc(n)
    local x,j ;
    mul( x-ithprime(n+j),j=0..4) ;
    expand(%) ;
    coeff(%,x,1) ;
end proc:
seq(A127489(n),n=1..60) ; # R. J. Mathar, Apr 23 2023
  • Mathematica
    Table[CoefficientList[Expand[(x-Prime[n])*(x-Prime[n+1])*(x-Prime[n+2])* (x-Prime[n+3])*(x-Prime[n+4])],x][[2]],{n,1,24}]

Extensions

Edited by Stefan Steinerberger, Jul 18 2007

A127491 Primes which are half of the absolute coefficients [x^2] of the 5th-order polynomials with prime roots as defined in A127489.

Original entry on oeis.org

310733, 426871, 15722159, 166492163, 177861107, 270396557, 342955763, 406947461, 1606837039, 1908243773, 2902193117, 3386269021, 5441167877, 6953015807, 7671152921, 10005413687, 10979785673, 14774655421, 16546239937
Offset: 1

Views

Author

Artur Jasinski, Jan 16 2007

Keywords

Comments

The polynomials are of the form (x-prime(i))*(x-prime(i+1))*..*(x-prime(i+4)). The quadratic terms have coefficients which are of the form -sum_{j

Examples

			The first contribution is from the 11th polynomial, (x-prime(11)) *(x-prime(12)) *(x-prime(13)) *(x-prime(14)) *(x-prime(15)) = x^5 -199x^4 +15766x^3 -621466x^2 +12185065x -95041567,
where the coefficient of [x^2] is -621466. Its sign-reversed half is 310733, a prime.

Crossrefs

Cf. A127345 - A127351, A006094, A046301 - A046303, A127489, A127490.

Programs

  • Maple
    isA127491 := proc(k)
    local x,j,p ;
    mul( x-ithprime(k+j),j=0..4) ;
    expand(%) ;
    abs(coeff(%,x,2)/2) ;
    isprime(%)
end proc:
A127491k := proc(n)
    option remember ;
    if n = 0 then
        0;
    else
        for k from procname(n-1)+1 do
            if isA127491(k) then
                return k ;
            end if;
        end do:
    end if;
end proc:
A127491 := proc(n)
    option remember ;
    local k ;
    k := A127491k(n) ;
    mul( x-ithprime(k+j),j=0..4) ;
    expand(%) ;
    abs(coeff(%,x,2)/2) ;
end proc:
seq(A127491(n),n=1..60) ; # R. J. Mathar, Apr 23 2023

Extensions

Entries replaced to comply with the definition. - R. J. Mathar, Sep 26 2011

A127492 Indices m of primes such that Sum_{k=0..2, k

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Jan 16 2007

Keywords

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

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.

Programs

  • 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 *)

Extensions

Definition simplified by R. J. Mathar, Apr 23 2023
Edited by Jon E. Schoenfield, Jul 23 2023
Showing 1-3 of 3 results.

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