A127349 -id:A127349 - 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

A127351 Prime numbers n such that A127350(k) = 2*n for some k.

Original entry on oeis.org

2003, 7883, 31151, 35363, 394739, 434939, 541007, 564983, 837929, 865979, 2453999, 2680493, 3479303, 3536219, 4145717, 4367267, 4706311, 5414159, 6541103, 6856019, 8804231, 9109223, 10227323, 10296059, 10701683, 10795507

Offset: 1

Artur Jasinski, Jan 11 2007

nonn

Primes of the form (Sum_{i=k..k+3}Sum_{j=i+1..k+4}prime(i)*prime(j))/2.

Primes of the form a/2 where a is the coefficient of x^3 of the polynomial Prod_{j=0,4}(x-prime(k+j)) for some k.

Cf. A127350, A127345, A127346, A127347, A127348, A127349, A070934, A006094.

a = {}; Do[If[PrimeQ[(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])/2], AppendTo[a, (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])/2]], {x, 1, 1000}]; a

{m=235;k=4;for(n=1,m,a=sum(i=n,n+k-1,sum(j=i+1,n+k,prime(i)*prime(j)));if(isprime(b=a/2),print1(b,",")))} \\ Klaus Brockhaus, Jan 21 2007

{m=235;k=4;for(n=1,m,a=polcoeff(prod(j=0,k,(x-prime(n+j))),3);if(isprime(b=a/2),print1(b,",")))} \\ Klaus Brockhaus, Jan 21 2007

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

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

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

A191216 Arithmetic derivative of prime(n) * prime(n+1) * prime(n+2) * prime(n+3) * prime(n+4) * prime(n+5).

Original entry on oeis.org

361, 230456, 1005768, 3462570, 11006128, 25925028, 61456764, 127697940, 249379116, 448408452, 740850012, 1263239320, 1914568816, 2884222410, 4371191782, 6287341056, 8758591370, 11640682466, 15938770638, 21721208748, 29153150298, 38784336168, 49888704100, 62506263054, 76188213990, 95511276660, 118760260290, 150724895476, 187405610004, 243040520764

Offset: 1

Giorgio Balzarotti, May 26 2011

nonn

Extension to 5 primes of concepts in A001043 A127489 A127349 A127345

Cf. A001043, A127489, A127349, A127345.

der:=n->n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
seq(dif(ithprime(i)*ithprime(i+1)*ithprime(i+2)*ithprime(i+3)*ithprime(i+4)*ithprime(i+5)),i=1..30);

a(n) = (prime(n) * prime(n+1) * prime(n+2) * prime(n+3) * prime(n+4) * prime(n+5))' where f' is the arithmetic derivative (see A003415) of f.

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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A127351 Prime numbers n such that A127350(k) = 2*n for some 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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A127492 Indices m of primes such that Sum_{k=0..2, k

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

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A191216 Arithmetic derivative of prime(n) * prime(n+1) * prime(n+2) * prime(n+3) * prime(n+4) * prime(n+5).

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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.