A046303 -id:A046303 - cp's OEIS frontend

A127343 Product of 11 consecutive primes.

200560490130, 3710369067405, 50708377254535, 436092044389001, 2928046583754721, 14107860812636383, 64027983688118969, 229747470880897477, 810162134158954261, 2500935283708076197

Offset: 1

Artur Jasinski, Jan 11 2007

nonn

a(n) is the absolute value of the coefficient of x^0 of the polynomial Product_{j=0..10} (x-prime(n+j)) of degree 11; the roots of this polynomial are prime(n), ..., prime(n+10).

Cf. A006094, A046301, A046302, A046303, A046324, A046325, A046326, A046327, A127342, A127344.

[&*[ NthPrime(n+k): k in [0..10] ]: n in [1..50] ]; // Vincenzo Librandi, Apr 03 2011

a = {}; Do[AppendTo[a, Product[Prime[x + n], {n, 0, 10}]], {x, 1, 50}]; a
Times@@@Partition[Prime[Range[50]],11,1] (* Harvey P. Dale, Oct 21 2011 *)

{m=10;k=11;for(n=0,m-1,print1(a=prod(j=1,k,prime(n+j)),","))} \\ Klaus Brockhaus, Jan 21 2007

{m=10;k=11;for(n=1,m,print1(abs(polcoeff(prod(j=0,k-1,(x-prime(n+j))),0)),","))} \\ Klaus Brockhaus, Jan 21 2007

Edited by Klaus Brockhaus, Jan 21 2007

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

Original entry on oeis.org

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

A098012 Triangle read by rows in which the k-th term in row n (n >= 1, k = 1..n) is Product_{i=0..k-1} prime(n-i).

Original entry on oeis.org

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

Offset: 1

Alford Arnold, Sep 09 2004

Also, square array A(m,n) in which row m lists all products of m consecutive primes (read by falling antidiagonals). See also A248164. - M. F. Hasler, May 03 2017

			2
3 3*2
5 5*3 5*3*2
7 7*5 7*5*3 7*5*3*2
Or, as an infinite square array:
     2     3     5     7  ... : row 1 = A000040,
     6    15    35    77  ... : row 2 = A006094,
    30   105   385  1001  ... : row 3 = A046301,
   210  1155  5005 17017  ... : row 4 = A046302,
   ..., with col.1 = A002110, col.2 = A070826, col.3 = A059865\{1}. - _M. F. Hasler_, May 03 2017

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A000040, A002110, A006094, A046301, A046302, A046303.

Cf. A060381 (central terms), A104887, A248147.

P:=Filtered([1..200],IsPrime);;
T:=Flat(List([1..9],n->List([1..n],k->Product([0..k-1],i->P[n-i])))); # Muniru A Asiru, Mar 16 2019

a098012 n k = a098012_tabl !! (n-1) !! (k-1)
a098012_row n = a098012_tabl !! (n-1)
a098012_tabl = map (scanl1 (*)) a104887_tabl
-- Reinhard Zumkeller, Oct 02 2014

T:=(n,k)->mul(ithprime(n-i),i=0..k-1): seq(seq(T(n,k),k=1..n),n=1..9); # Muniru A Asiru, Mar 16 2019

Flatten[ Table[ Product[ Prime[i], {i, n, j, -1}], {n, 9}, {j, n, 1, -1}]] (* Robert G. Wilson v, Sep 21 2004 *)

T098012(n,k)=prod(i=0,k-1,prime(n-i)) \\ "Triangle" variant
A098012(m,n)=prod(i=0,m-1,prime(n+i)) \\ "Square array" variant. - M. F. Hasler, May 03 2017

n-th row = partial products of row n in A104887. - Reinhard Zumkeller, Oct 02 2014

More terms from Robert G. Wilson v, Sep 21 2004

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

A175742 Numbers with 32 divisors.

Original entry on oeis.org

840, 1080, 1320, 1512, 1560, 1848, 1890, 1920, 2040, 2184, 2280, 2310, 2376, 2688, 2730, 2760, 2808, 2856, 2970, 3000, 3080, 3192, 3432, 3456, 3480, 3510, 3570, 3640, 3672, 3720, 3864, 3990, 4104, 4158, 4224, 4290, 4440, 4480, 4488, 4590, 4760, 4830, 4872

Offset: 1

Jaroslav Krizek, Aug 27 2010

nonn

Numbers of the form p^31, p^15*q^1, p^7*q^3, p^7*q^1*r^1, p^3*q^3*r^1, p^3*q^1*r^1*s^1 and p^1*q^1*r^1*s^1*t^1, where p, q, r, s and t are distinct primes.

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, Index entries for number of divisors

Cf. A137491, A137492, A137493, A139571.

Cf. A046303 (a subsequence). - Michel Marcus, Apr 06 2017

Select[Range[10000],DivisorSigma[0,#]==32&] (* Vladimir Joseph Stephan Orlovsky, May 06 2011 *)

is(n)=numdiv(n)==32 \\ Charles R Greathouse IV, Jun 19 2016

A000005(a(n))=32.

Extended by T. D. Noe, May 09 2011

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

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

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

A261210 a(n) = gpf(1 + Product_{k=0..4} prime(n+k)), where gpf is greatest prime factor and prime(i) is the i-th prime.

Original entry on oeis.org

2311, 1877, 163, 80831, 12647, 6967, 139, 3633983, 1657, 15473, 2970049, 933853, 64776587, 99767, 21067, 84961, 1524137, 820319, 157229, 489427, 2066140207, 71899, 15287, 1680583, 769117, 55732973, 52889, 225941, 4678959379, 1491686591, 87701

Offset: 1

Anders Hellström, Aug 12 2015

nonn

Anders Hellström, Table of n, a(n) for n = 1..1000

Cf. A006530, A046303.

Array[FactorInteger[1 + Product[Prime[# + k], {k, 0, 4}]][[-1, 1]] &, {31}] (* Michael De Vlieger, Aug 19 2015 *)

gpf(n)=vecmax(factor(n)[, 1]);
first(m)=vector(m, i, gpf(1+prod(j=0,4,prime(i+j))));

a(n) = A006530(1+A046303(n)). - Michel Marcus, Aug 13 2015

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A127343 Product of 11 consecutive primes.

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

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A098012 Triangle read by rows in which the k-th term in row n (n >= 1, k = 1..n) is Product_{i=0..k-1} prime(n-i).

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A127342 Product of 10 consecutive primes.

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A175742 Numbers with 32 divisors.

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A127344 Product of 12 consecutive primes.

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A127491 Primes which are half of the absolute coefficients [x^2] of the 5th-order polynomials with prime roots as defined in A127489.