A127351 -id:A127351 - cp's OEIS frontend

A127345 a(n) = pq + pr + qr with p = prime(n), q = prime(n+1), and r = prime(n+2).

31, 71, 167, 311, 551, 791, 1151, 1655, 2279, 3119, 3935, 4871, 5711, 6791, 8391, 9959, 11639, 13175, 14831, 16559, 18383, 20975, 24071, 27419, 30191, 32231, 33911, 36071, 40511, 45791, 51983, 55199, 60167, 64199, 69599, 73911, 79031, 84311

Offset: 1

Artur Jasinski, Jan 11 2007

a(n) = coefficient of x^1 of the polynomial Product_{j=0..2} (x-prime(n+j)) of degree 3; the roots of this polynomial are prime(n), ..., prime(n+2); cf. Vieta's formulas.

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

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Vieta's Formulas

Cf. A127346, A127347, A127348, A127349, A127350, A127351, A070934, A006094.

Table[Prime[n]*Prime[n+1] + Prime[n]*Prime[n+2] + Prime[n+1]*Prime[n+2], {n, 100}]
Total[Times@@@Subsets[#,{2}]]&/@Partition[Prime[Range[40]],3,1] (* Harvey P. Dale, Sep 11 2017 *)

{m=38;k=2;for(n=1,m,print1(sum(i=n,n+k-1,sum(j=i+1,n+k,prime(i)*prime(j))),","))} \\ Klaus Brockhaus, Jan 21 2007

{m=38;k=2;for(n=1,m,print1(polcoeff(prod(j=0,k,(x-prime(n+j))),1),","))} \\ Klaus Brockhaus, Jan 21 2007

p=2;q=3;forprime(r=5,1e3,print1(p*q+p*r+q*r", ");p=q;q=r) \\ Charles R Greathouse IV, Jan 13 2012

Edited by Klaus Brockhaus, Jan 21 2007

A127346 Primes in A127345.

Original entry on oeis.org

31, 71, 167, 311, 1151, 3119, 4871, 5711, 6791, 14831, 24071, 33911, 60167, 79031, 101159, 106367, 115631, 158231, 235751, 259751, 366791, 402551, 455471, 565919, 635711, 644951, 1124831, 1347971, 1510799, 1547927, 1743419, 1851671, 2048471

Offset: 1

Artur Jasinski, Jan 11 2007

nonn

Primes of the form prime(k)*prime(k+1) + prime(k)*prime(k+2) + prime(k+1)*prime(k+2).

A prime number n is in the sequence if for some k it is the coefficient of x^1 of the polynomial Product_{j=0..2} (x-prime(k+j)); the roots of this polynomial are prime(k), ..., prime(k+2).

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A127345, A127347, A127351, A006094, A002110, A034962, A034965, A082246, A082251, A127340, A127341, A070934, A046301, A046302, A046303, A046324, A046325, A046326, A046327.

b = {}; a = {}; Do[If[PrimeQ[Prime[x] Prime[x + 1] + Prime[x] Prime[x + 2] + Prime[x + 1] Prime[x + 2]], AppendTo[a, Prime[x] Prime[x + 1] + Prime[x] Prime[x + 2] + Prime[x + 1] Prime[x + 2]], AppendTo[b, Prime[x] Prime[x + 1] + Prime[x] Prime[x + 2] + Prime[x + 1] Prime[x + 2]]], {x, 1, 100}]; Print[a] (* Artur Jasinski, Jan 11 2007 *)
s[li_] := li[[1]]*(li[[2]]+li[[3]])+li[[2]]*li[[3]]; Select[(s[#]&/@Partition[Prime[Range[100]], 3, 1]), PrimeQ] (* Zak Seidov, Jan 13 2012 *)

{m=143;k=2;for(n=1,m,a=sum(i=n,n+k-1,sum(j=i+1,n+k,prime(i)*prime(j)));if(isprime(a),print1(a,",")))} \\ Klaus Brockhaus, Jan 21 2007

{m=143;k=2;for(n=1,m,a=polcoeff(prod(j=0,k,(x-prime(n+j))),1);if(isprime(a),print1(a,",")))} \\ Klaus Brockhaus, Jan 21 2007

p=2; q=3; forprime(r=5, 1e3, if(isprime(t=p*q+p*r+q*r), print1(t", ")); p=q; q=r) \\ Charles R Greathouse IV, Jan 13 2012

a(n) = A127345(A204231(n)). - Zak Seidov, Jan 13 2012

Edited and extended by Klaus Brockhaus, Jan 21 2007

A127349 a(n) = Sum_{i=n..n+1} Sum_{j=i+1..n+2} Sum_{k=j+1..n+3} prime(i)prime(j)prime(k).

Original entry on oeis.org

247, 886, 2556, 6288, 12900, 22392, 40808, 63978, 105000, 161142, 216232, 294168, 385544, 507782, 658820, 858000, 1067502, 1251952, 1518910, 1783854, 2114748, 2618148, 3147710, 3696090, 4239528, 4626300, 5033232, 5898936, 6871200

Offset: 1

Artur Jasinski, Jan 11 2007

nonn

a(n) = absolute value of the coefficient of x^1 of the polynomial Product_{j=0..3} (x - prime(n+j)) of degree 4; the roots of this polynomial are prime(n), ..., prime(n+3); cf. Vieta's formulas.
All terms with exception of the first one are even.
Arithmetic derivative (see A003415) of prime(n)*prime(n+1)*prime(n+2)*prime(n+3). - Giorgio Balzarotti, May 26 2011

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Vieta's Formulas

Cf. A046302, A127345, A127346, A127347, A127348, A127350, A127351, A070934, A006094.

[NthPrime(n)*NthPrime(n+1)*NthPrime(n+2) + NthPrime(n)*NthPrime(n+2)*NthPrime(n+3) + NthPrime(n)*NthPrime(n+1)* NthPrime(n+3) + NthPrime(n+1)*NthPrime(n+2)*NthPrime(n+3): n in [1..30]]; // Vincenzo Librandi, Feb 12 2018

P := select(isprime, [2, seq(i, i = 1 .. 1000, 2)]):
f := L) -> convert(L, `*`)*add(1/t, t = L):
seq(f(P[i..i+3]),i=1..nops(P)-3); # Robert Israel, Feb 11 2018

Table[Prime[n] Prime[n+1] Prime[n+2] + Prime[n] Prime[n+2] Prime[n+3] + Prime[n] Prime[n+1] Prime[n+3] + Prime[n+1] Prime[n+2] Prime[n+3], {n, 100}]

{m=29;h=3;for(n=1,m,print1(sum(i=n,n+h-2,sum(j=i+1,n+h-1,sum(k=j+1,n+h,prime(i)*prime(j)*prime(k)))),","))} \\ Klaus Brockhaus, Jan 21 2007

{m=29;k=3;for(n=1,m,print1(abs(polcoeff(prod(j=0,k,(x-prime(n+j))),1)),","))} \\ Klaus Brockhaus, Jan 21 2007

a(n) = A046302(n)*Sum_{i=n..n+3} 1/prime(i). - Robert Israel, Feb 11 2018

Edited by Klaus Brockhaus, Jan 21 2007

A127348 Coefficient of x^2 in the polynomial (x-p(n))(x-p(n+1))(x-p(n+2))*(x-p(n+3)), where p(k) is the k-th prime.

Original entry on oeis.org

101, 236, 466, 838, 1330, 1918, 2862, 3856, 5350, 7096, 8622, 10558, 12654, 15228, 18090, 21550, 24916, 27702, 31500, 35068, 39298, 45322, 51240, 56980, 62398, 66130, 69958, 77854, 86230, 96618, 106888, 115842, 124342, 133122, 144090, 152568, 163282, 174348

Offset: 1

Artur Jasinski, Jan 11 2007

nonn

			a(1)=101 because (x-2)*(x-3)*(x-5)*(x-7) = x^4 - 17x^3 + 101x^2 - 247x + 210.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Vieta's Formulas

Cf. A127345, A127346, A127347, A127349, A127350, A127351, A070934, A006094.

a:=n->coeff(expand((x-ithprime(n))*(x-ithprime(n+1))*(x-ithprime(n+2))*(x-ithprime(n+3))),x,2): seq(a(n),n=1..45); # Emeric Deutsch, Jan 20 2007

Table[Prime[x] Prime[x + 1] + Prime[x] Prime[x + 2] + Prime[x] Prime[x + 3] + Prime[x + 1] Prime[x + 2] + Prime[x + 1] Prime[x + 3] + Prime[x + 2] Prime[x + 3], {x, 1, 100}]
Total[Times@@@Subsets[#,{2}]]&/@Partition[Prime[Range[40]],4,1] (* Harvey P. Dale, Apr 15 2019 *)

{m=35;k=3;for(n=1,m,print1(sum(i=n,n+k-1,sum(j=i+1,n+k,prime(i)*prime(j))),","))} \\ Klaus Brockhaus, Jan 21 2007

{m=35;k=3;for(n=1,m,print1(abs(polcoeff(prod(j=0,k,(x-prime(n+j))),2)),","))} \\ Klaus Brockhaus, Jan 21 2007

a(n) = p(n)*p(n+1) + p(n)*p(n+2) + p(n)*p(n+3) + p(n+1)*p(n+2) + p(n+1)*p(n+3) + p(n+2)*p(n+3), where p(k) is the k-th prime (by Viete's formula relating the zeros and the coefficients of a polynomial). - Emeric Deutsch, Jan 20 2007

Edited by Emeric Deutsch and Klaus Brockhaus, Jan 20 2007

A127350 a(n) = Sum_{i=n..n+3} Sum_{j=i+1..n+4} prime(i)*prime(j).

Original entry on oeis.org

288, 574, 1078, 1750, 2710, 4006, 5590, 7630, 10270, 13030, 15766, 19462, 23510, 27550, 32830, 38590, 43750, 49190, 55570, 62302, 70726, 80470, 89350, 98710, 106870, 113590, 124822, 137590, 151990, 167230, 186454, 199798, 214774, 230270, 247630, 262942, 281422

Offset: 1

Artur Jasinski, Jan 11 2007

nonn

a(n) = absolute value of the coefficient of x^3 of the polynomial Prod_{j=0,4}(x-prime(n+j)) of degree 5; the roots of this polynomial are prime(n), ..., prime(n+4); cf. Vieta's formulas.

All terms are even.

Eric Weisstein's World of Mathematics, Vieta's Formulas

Cf. A000040, A127345, A127346, A127347, A127348, A127349, A127351, A070934, A006094.

Table[Prime[x] Prime[x + 1] + Prime[x] Prime[x + 2] + Prime[x] Prime[x + 3] + Prime[x] Prime[x + 4] + Prime[x + 1] Prime[x + 2] + Prime[x + 1] Prime[x + 3] + Prime[x + 1] Prime[x + 4] + Prime[x + 2] Prime[x + 3] + Prime[x + 2] Prime[x + 4] + Prime[x + 3] Prime[x + 4], {x, 1, 100}]

{m=34;k=4;for(n=1,m,print1(sum(i=n,n+k-1,sum(j=i+1,n+k,prime(i)*prime(j))),","))} \\ Klaus Brockhaus, Jan 21 2007

{m=34;k=4;for(n=1,m,print1(abs(polcoeff(prod(j=0,k,(x-prime(n+j))),3)),","))} \\ Klaus Brockhaus, Jan 21 2007

Edited by Klaus Brockhaus, Jan 21 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

Artur Jasinski, Jan 16 2007

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

			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.

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

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

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

Artur Jasinski, Jan 16 2007

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 = 2..1000

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.

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 *)

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

Edited by Jon E. Schoenfield, Jul 23 2023

A127493 Indices k such that the coefficient [x^1] of the polynomial Product_{j=0..4} (x-prime(k+j)) is prime.

Original entry on oeis.org

1, 5, 8, 9, 22, 29, 45, 49, 60, 69, 87, 89, 90, 107, 114, 124, 125, 131, 134, 138, 145, 156, 171, 183, 188, 191, 203, 204, 207, 212, 219, 255, 261, 290, 298, 303, 329, 330, 343, 344, 349, 354, 378, 397, 398, 400, 403, 454, 456, 466, 474, 515, 549, 560, 570, 578

Offset: 1

Artur Jasinski, Jan 16 2007

nonn

A fifth-order polynomial with 5 roots which are the five consecutive primes from prime(k) onward is defined by Product_{j=0..4} (x-prime(k+j)). The sequence is a catalog of the cases where the coefficient of its linear term is prime.

Indices k such that e4(prime(k), prime(k+1), ..., prime(k+4)) is prime, where e4 is the elementary symmetric polynomial summing all products of four variables. - Charles R Greathouse IV, Jun 15 2015

			For k=2, the polynomial is (x-3)*(x-5)*(x-7)*(x-11)*(x-13) = x^5-39*x^4+574*x^3-3954*x^2+12673*x-15015, where 12673 is not prime, so k=2 is not in the sequence.
For k=5, the polynomial is x^5-83*x^4+2710*x^3-43490*x^2+342889*x-1062347, where 342889 is prime, so k=5 is in the sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001043, A034961, A034963, A034964, A127333-A127343, A127345-A127351, A037171, A034962, A034965, A082246, A082251, A070934, A006094, A046301-A046303, A046324-A046327, A127489, A127491, A127492, A024449.

isA127493 := proc(k)
    local x,j ;
    mul( x-ithprime(k+j),j=0..4) ;
    expand(%) ;
    isprime(coeff(%,x,1)) ;
end proc:
A127493 := proc(n)
    option remember ;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA127493(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A127493(n),n=1..60) ; # R. J. Mathar, Apr 23 2023

a = {}; Do[If[PrimeQ[(Prime[x] Prime[x + 1]Prime[x + 2]Prime[x + 3] + Prime[x] Prime[x + 2]Prime[x + 3]Prime[x + 4] + Prime[x] Prime[x + 1]Prime[x + 3]Prime[x + 4] + Prime[x] Prime[x + 1]Prime[x + 2]Prime[x + 4] + Prime[x + 1] Prime[x + 2]Prime[x + 3]Prime[x + 4])], AppendTo[a, x]], {x, 1, 1000}]; a

e4(v)=sum(i=1,#v-3, v[i]*sum(j=i+1,#v-2, v[j]*sum(k=j+1,#v-1, v[k]*vecsum(v[k+1..#v]))))
pr(p, n)=my(v=vector(n)); v[1]=p; for(i=2,#v, v[i]=nextprime(v[i-1]+1)); v
is(n,p=prime(n))=isprime(e4(pr(p,5)))
v=List(); n=0; forprime(p=2,1e4, if(is(n++,p), listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Jun 15 2015

Definition and comment rephrased and examples added by R. J. Mathar, Oct 01 2009

A125270 Coefficient of x^2 in polynomial whose zeros are 5 consecutive primes starting with the n-th prime.

Original entry on oeis.org

1358, 3954, 10478, 22210, 43490, 78014, 129530, 206650, 324350, 466270, 621466, 853742, 1132130, 1436690, 1870850, 2388050, 2886370, 3440410, 4133410, 4904906, 5926654, 7195670, 8425430, 9792950, 11040910, 12098990, 13917898, 16097810

Offset: 1

Artur Jasinski, Jan 16 2007

nonn

Sums of all distinct products of 3 out of 5 consecutive primes, starting with the n-th prime; value of 3rd elementary symmetric function on the 5 consecutive primes.

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.

a = {}; Do[AppendTo[a, (Prime[x] Prime[x + 1] Prime[x + 2] + Prime[x] Prime[x + 1] Prime[x + 3] + Prime[x] Prime[x + 1] Prime[x + 4] + Prime[x] Prime[x + 2] Prime[x + 3] + Prime[x] Prime[x + 2] Prime[x + 4] + Prime[x] Prime[x + 3] Prime[x + 4] + Prime[x + 1] Prime[x + 2] Prime[x + 3] + Prime[x + 1] Prime[x + 2] Prime[x + 4] + Prime[x + 1] Prime[x + 3] Prime[x + 4] + Prime[x + 2] Prime[x + 3] Prime[x + 4])], {x, 1, 100}]; a
fcp[{p_,q_,r_,s_,t_}]:=p*q(r+s+t)+(p+q)r(s+t)+(p+q+r)s*t; fcp/@Partition[ Prime[ Range[40]],5,1] (* Harvey P. Dale, Sep 05 2014 *)

Let p = Prime(n), q = Prime(n+1), r = Prime(n+2), s = Prime(n+3) and t = Prime(n+4). Then a(n) = p q (r+s+t) + (p + q) r (s + t) + (p + q + r) s t.

Edited and corrected by Franklin T. Adams-Watters, Jan 23 2007

Showing 1-9 of 9 results.

cp's OEIS Frontend

A127345 a(n) = pq + pr + qr with p = prime(n), q = prime(n+1), and r = prime(n+2).

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A127346 Primes in A127345.

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A127349 a(n) = Sum_{i=n..n+1} Sum_{j=i+1..n+2} Sum_{k=j+1..n+3} prime(i)*prime(j)*prime(k).

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A127348 Coefficient of x^2 in the polynomial (x-p(n))*(x-p(n+1))*(x-p(n+2))*(x-p(n+3)), where p(k) is the k-th prime.

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A127350 a(n) = Sum_{i=n..n+3} Sum_{j=i+1..n+4} prime(i)*prime(j).

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

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

A127349 a(n) = Sum_{i=n..n+1} Sum_{j=i+1..n+2} Sum_{k=j+1..n+3} prime(i)prime(j)prime(k).

A127348 Coefficient of x^2 in the polynomial (x-p(n))(x-p(n+1))(x-p(n+2))*(x-p(n+3)), where p(k) is the k-th prime.