A127345 -id:A127345 - cp's OEIS frontend

A127346 Primes in A127345.

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

A127347 Composites in A127345.

Original entry on oeis.org

551, 791, 1655, 2279, 3935, 8391, 9959, 11639, 13175, 16559, 18383, 20975, 27419, 30191, 32231, 36071, 40511, 45791, 51983, 55199, 64199, 69599, 73911, 84311, 89751, 94679, 112511, 122759, 133419, 145571, 153671, 163775, 169439, 178079

Offset: 1

Artur Jasinski, Jan 11 2007

nonn

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

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

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A127345, A127346, A127347.

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]; Print[b]
Select[Total[Times@@@Subsets[#,{2}]]&/@Partition[Prime[ Range[80]], 3,1],!PrimeQ[#]&] (* Harvey P. Dale, May 27 2012 *)

{m=52;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=52;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

Edited and extended by Klaus Brockhaus, Jan 21 2007

A204231 Position of primes in A127345.

Original entry on oeis.org

1, 2, 3, 4, 7, 10, 12, 13, 14, 19, 23, 27, 33, 37, 41, 42, 44, 49, 59, 61, 69, 72, 76, 83, 88, 89, 111, 121, 126, 127, 134, 137, 143, 144, 146, 149, 159, 163, 170, 177, 178, 186, 189, 195, 197, 198, 204, 208, 214, 217, 220, 224, 228, 233, 234, 236, 243, 247, 248, 249, 276, 278, 288, 294, 295, 335, 338, 353, 354, 380, 382, 384, 395, 401, 402, 408, 411, 427

Offset: 1

Zak Seidov, Jan 13 2012

nonn

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

Cf. A127345, A127346.

p=2; q=3; n=0; forprime(r=5, 1e3, n++; if(isprime(p*q+p*r+q*r), print1(n", ")); p=q; q=r) \\ Charles R Greathouse IV, Jan 13 2012

A127346(n) = A127345(a(n)).

A287609 Intersection of A034961 and A127345.

Original entry on oeis.org

31, 71, 311, 551, 1151, 14831, 45791, 455471, 2035271, 6345239, 7241615, 8290031, 8329991, 9086231, 9324351, 10449575, 11497199, 15454151, 16515815, 18337271, 20650811, 22946591, 27609311, 33220079, 40487471, 44106191, 45015791, 49021199, 53315519, 54536519

Offset: 1

Zak Seidov, May 27 2017

nonn

Surprisingly many terms are prime numbers: 31,71,311,1151,14831,455471.
Positions of a(n) in A127345: {1,2,4,5,7,19,30,76,142}.
Positions of a(n) in A034961: {4,8,26,41,75,660,1780,14009,54929}.
Positions of primes in a(n): {1,2,3,5,6,8,21,22,25,32,37,39,40,45,49,50, 59,62,66,69,...}. - Michael De Vlieger, May 28 2017

			31 is in the sequence because it is both the total of three consecutive primes (7 + 11 + 13) and it is (2*3 + 2*5 + 3*5) = (6 + 10 + 15). - _Michael De Vlieger_, May 28 2017

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A034961, A127345.

Intersection[Map[Total, #], Map[#1 #2 + #1 #3 + #2 #3 & @@ # &, #]] &@ Partition[Prime@ Range[10^6], 3, 1] (* Michael De Vlieger, May 28 2017 *)

from _future_ import division
from sympy import isprime, prevprime, nextprime
A287609_list, p, q, r = [], 2, 3, 5
while r < 10**6:
    n = p*(q+r) + q*r
    m = n//3
    pm, nm = prevprime(m), nextprime(m)
    k = n - pm - nm
    if isprime(m):
        if m == k:
            A287609_list.append(n)
    else:
        if nextprime(nm) == k or prevprime(pm) == k:
            A287609_list.append(n)
    p, q, r = q, r, nextprime(r) # Chai Wah Wu, May 31 2017

More terms from Michael De Vlieger, May 28 2017

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

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

Showing 1-10 of 15 results. Next

cp's OEIS Frontend

A127346 Primes in A127345.

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A127347 Composites in A127345.

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A204231 Position of primes in A127345.

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A287609 Intersection of A034961 and A127345.

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

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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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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.