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 1-9 of 9 results.

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

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Jan 11 2007

Keywords

Comments

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]

Crossrefs

Programs

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

Extensions

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

Views

Author

Artur Jasinski, Jan 11 2007

Keywords

Comments

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).

Crossrefs

Programs

  • Mathematica
    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 *)
  • PARI
    {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
    
  • PARI
    {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
    
  • PARI
    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

Formula

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

Extensions

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

Views

Author

Artur Jasinski, Jan 11 2007

Keywords

Comments

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

Crossrefs

Programs

  • Magma
    [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
  • Maple
    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
  • Mathematica
    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}]
  • PARI
    {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
    
  • PARI
    {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
    

Formula

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

Extensions

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

Views

Author

Artur Jasinski, Jan 11 2007

Keywords

Examples

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

Crossrefs

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    {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
    
  • PARI
    {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

Formula

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

Extensions

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

Views

Author

Artur Jasinski, Jan 11 2007

Keywords

Comments

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.

Crossrefs

Programs

  • Mathematica
    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}]
  • PARI
    {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
    
  • PARI
    {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

Extensions

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

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.
		

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

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

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

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

Author

Artur Jasinski, Jan 16 2007

Keywords

Comments

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

Examples

			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.
		

Programs

  • Maple
    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
  • Mathematica
    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
  • PARI
    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

Extensions

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

Author

Artur Jasinski, Jan 16 2007

Keywords

Comments

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.

Programs

  • Mathematica
    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 *)

Formula

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.

Extensions

Edited and corrected by Franklin T. Adams-Watters, Jan 23 2007
Showing 1-9 of 9 results.