A127343 -id:A127343 - cp's OEIS frontend

A096334 Triangle read by rows: T(n,k) = prime(n)#/prime(k)#, 0<=k<=n.

1, 2, 1, 6, 3, 1, 30, 15, 5, 1, 210, 105, 35, 7, 1, 2310, 1155, 385, 77, 11, 1, 30030, 15015, 5005, 1001, 143, 13, 1, 510510, 255255, 85085, 17017, 2431, 221, 17, 1, 9699690, 4849845, 1616615, 323323, 46189, 4199, 323, 19, 1, 223092870, 111546435, 37182145, 7436429, 1062347, 96577, 7429, 437, 23, 1

Offset: 0

Reinhard Zumkeller, Aug 03 2004

T(n,k) is the (k+1)-th product of (n-k) successive primes (k, n-(k+1) >= 0). - Alois P. Heinz, Jan 21 2022

			Triangle begins:
    1;
    2,   1;
    6,   3,  1;
   30,  15,  5, 1;
  210, 105, 35, 7, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

T(n-1+j,n-1) give (j=1-12): A000040, A006094, A046301, A046302, A046303, A046324, A046325, A046326, A046327, A127342, A127343, A127344.
Columns k=0-1 give: A002110, A070826.
T(2n,n) gives A107712.
Row sums give A350895.
Antidiagonal sums give A350758.
Cf. A073485 (distinct values sorted).

T:= proc(n, k) option remember;
     `if`(n=k, 1, T(n-1, k)*ithprime(n))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jan 21 2022

T[n_, k_] := Times @@ Prime[Range[k + 1, n]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 13 2021 *)

pr(n) = factorback(primes(n)); \\ A002110
row(n) = my(P=pr(n)); vector(n+1, k, P/pr(k-1)); \\ Michel Marcus, Jan 21 2022

T(n,0) = A002110(n); T(n,n) = 1;
T(n,n-1) = A000040(n) for n>0;
T(n,k) = A002110(n)/A002110(k), 0<=k<=n.
T(n,k) = Product_{j=k+1..n} prime(j). - Alois P. Heinz, Jan 21 2022

A127342 Product of 10 consecutive primes.

Original entry on oeis.org

6469693230, 100280245065, 1236789689135, 10141675450907, 62298863484143, 266186053068611, 1085220062510491, 3766351981654057, 12091972151626183, 35224440615606707, 86239147714071593, 203079283326684719

Offset: 1

Artur Jasinski, Jan 11 2007

nonn

a(n) = coefficient of x^0 of the polynomial Product_{j=0..9} (x-prime(n+j)) of degree 10; the roots of this polynomial are prime(n), ..., prime(n+9).

Cf. A006094, A046301, A046302, A046303, A046324, A046325, A046326, A046327, A127343, A127344.

[&*[ NthPrime(n+k): k in [0..9] ]: n in [1..50] ]; // Vincenzo Librandi, Apr 03 2011

a = {}; Do[AppendTo[a, Product[Prime[x + n], {n, 0, 9}]], {x, 1, 50}]; a
Times@@@Partition[Prime[Range[50]],10,1] (* Harvey P. Dale, Oct 21 2011 *)

{m=12;k=10;for(n=0,m-1,print1(a=prod(j=1,k,prime(n+j)),","))} \\ Klaus Brockhaus, Jan 21 2007

{m=12;k=10;for(n=1,m,print1(polcoeff(prod(j=0,k-1,(x-prime(n+j))),0),","))} \\ Klaus Brockhaus, Jan 21 2007

Edited by Klaus Brockhaus, Jan 21 2007

A127344 Product of 12 consecutive primes.

Original entry on oeis.org

7420738134810, 152125131763605, 2180460221945005, 20496326086283047, 155186468939000213, 832363787945546597, 3905707004975257109, 15393080549020130959, 57521511525285752531, 182568275710689562381, 497341164867050876831, 1331590860773071702483

Offset: 1

Artur Jasinski, Jan 11 2007

nonn

a(n) = coefficient of x^0 of the polynomial Prod_{j=0,11}(x-prime(n+j)) of degree 12; the roots of this polynomial are prime(n), ..., prime(n+11).

Cf. A006094, A046301, A046302, A046303, A046324, A046325, A046326, A046327, A127342, A127343.

[&*[ NthPrime(n+k): k in [0..11] ]: n in [1..50] ]; // Vincenzo Librandi, Apr 03 2011

A127344 := proc(n) mul(ithprime(n+k),k=0..11) ; end proc: # R. J. Mathar, Apr 05 2011

a = {}; Do[AppendTo[a, Product[Prime[x + n], {n, 0, 11}]], {x, 1, 50}]; a
Times@@@Partition[Prime[Range[50]],12,1] (* Harvey P. Dale, Oct 21 2011 *)

{m=10;k=12;for(n=0,m-1,print1(a=prod(j=1,k,prime(n+j)),","))} \\ Klaus Brockhaus, Jan 21 2007

{m=10;k=12;for(n=1,m,print1(polcoeff(prod(j=0,k-1,(x-prime(n+j))),0),","))} \\ 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

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

cp's OEIS Frontend

A096334 Triangle read by rows: T(n,k) = prime(n)#/prime(k)#, 0<=k<=n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A127342 Product of 10 consecutive primes.

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Extensions

A127344 Product of 12 consecutive primes.

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

Extensions