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.

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A331259 Numerator of harmonic mean of 3 consecutive primes. Denominators are A331260.

Original entry on oeis.org

90, 315, 1155, 3003, 7293, 12597, 22287, 38019, 62031, 99789, 141081, 195693, 248583, 321339, 146969, 572241, 723399, 870531, 1041783, 1228371, 1435983, 1750719, 2149617, 2615799, 3027273, 3339363, 3603867, 3953757, 4692777, 5639943, 6837807, 7483899, 8512221
Offset: 1

Views

Author

Hugo Pfoertner, Jan 19 2020

Keywords

Examples

			b(1) = a(1)/A331260(1) = 3*2*3*5 / (3*5 + 2*5 + 2*3) = 90/31,
b(2) = a(2)/A331260(2) = 3*3*5*7 / (5*7 + 3*7 + 3*5) = 315/71,
...
b(15) = a(15)/A331260(15) = 3*47*53*59 / (53*59 + 47*59 + 47*53) = 440907/8391 = 146969/2797. The common factor of 3 (see A292530) makes the denominator different from A127345(15).

Links

Crossrefs

Cf. A046301, A127345, A292530, A331260.

Programs

  • Maple
    q:= proc(a,b,c) if nops({a,b,c} mod 3) = 1 then a*b*c else 3*a*b*c fi end proc:
P:= [seq(ithprime(i),i=1..102)]:
seq(q(P[i],P[i+1],P[i+2]),i=1..100); # Robert Israel, Jul 29 2024
  • PARI
    hm3(x,y,z)=3/(1/x+1/y+1/z);
p1=2; p2=3; forprime(p3=5,150, print1(numerator(hm3(p1,p2,p3)),", ");p1=p2;p2=p3)

Formula

a(n) = numerator ((3*p1*p2*p3)/(p2*p3 + p1*p3 + p1*p2)) with p1 = prime(n), p2 = prime(n + 1), p3 = prime(n + 2).
a(n) = p1*p2*p3 if p1 == p2 == p3 (mod 3), otherwise 3*p1*p2*p3. - Robert Israel, Jul 29 2024

A331260 Denominator of harmonic mean of 3 consecutive primes. Numerators are A331259.

Original entry on oeis.org

31, 71, 167, 311, 551, 791, 1151, 1655, 2279, 3119, 3935, 4871, 5711, 6791, 2797, 9959, 11639, 13175, 14831, 16559, 18383, 20975, 24071, 27419, 30191, 32231, 33911, 36071, 40511, 45791, 51983, 55199, 60167, 64199, 69599, 24637, 79031, 84311, 29917, 94679
Offset: 1

Views

Author

Hugo Pfoertner, Jan 19 2020

Keywords

Examples

			See A331259.

Links

Crossrefs

Cf. A046301, A127345, A292530, A331259.

Programs

  • Maple
    P:= [seq(ithprime(i),i=1..102)]:
f:= proc(a,b,c) if nops({a,b,c} mod 3) = 1 then (a*b+a*c+b*c)/3 else a*b+a*c+b*c fi end proc;
[seq(f(P[i],P[i+1],P[i+2]),i=1..100)]; # Robert Israel, Jul 28 2024
  • PARI
    hm3(x, y, z)=3/(1/x+1/y+1/z);
p1=2; p2=3; forprime(p3=5, 190, print1(denominator(hm3(p1, p2, p3)), ", "); p1=p2; p2=p3)

Formula

a(n) = denominator ((3*p1*p2*p3)/(p2*p3 + p1*p3 + p1*p2)) with p1 = prime(n), p2 = prime(n + 1), p3 = prime(n + 2).
a(n) = (p1*p2 + p1*p3 + p2*p3)/3 if p1 == p2 == p3 (mod 3), otherwise p1*p2 + p1*p3 + p2*p3. - Robert Israel, Jul 29 2024

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

Views

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.

Links

Crossrefs

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

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

Views

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.

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.

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

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

Views

Author

Giorgio Balzarotti, May 26 2011

Keywords

Links

  • Extension to 5 primes of concepts in A001043 A127489 A127349 A127345

Crossrefs

Cf. A001043, A127489, A127349, A127345.

Programs

  • Maple
    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);

Formula

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.
Previous Showing 11-15 of 15 results.

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