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 1-5 of 5 results.

A260613 Triangle read by rows: T(n, k) = coefficient of x^(n-k) in Product_{m=1..n} (x+prime(m)); 0 <= k <= n, n >= 0.

Original entry on oeis.org

1, 1, 2, 1, 5, 6, 1, 10, 31, 30, 1, 17, 101, 247, 210, 1, 28, 288, 1358, 2927, 2310, 1, 41, 652, 5102, 20581, 40361, 30030, 1, 58, 1349, 16186, 107315, 390238, 716167, 510510, 1, 77, 2451, 41817, 414849, 2429223, 8130689, 14117683, 9699690
Offset: 0

Views

Author

Matthew Campbell, Aug 10 2015

Keywords

Comments

Up to signs and order of coefficients the same as A070918. Except for signs and the first column the same as A238146. - M. F. Hasler, Aug 13 2015

Examples

			The triangle starts:
Row 0: 1;
Row 1: 1, 2;  Coefficients of x + 2.
Row 2: 1, 5, 6;  Coefficients of (x+2)(x+3) = x^2 + 5x + 6.
Row 3: 1, 10, 31, 30; Coeff's of (x+2)(x+3)(x+5) = x^3 + 10x^2 + 31x + 30.
Row 5: 1, 17, 101, 247, 210;
Row 6: 1, 28, 288, 1358, 2927, 2310;
...

Links

Crossrefs

Main diagonal gives A002110.
Row sums give A054640.
Cf. A000040.

Programs

  • Maple
    T:= n-> (p-> seq(coeff(p, x, n-i), i=0..n))(mul(x+ithprime(i), i=1..n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Aug 18 2019
  • Mathematica
    row[n_] := CoefficientList[Product[x + Prime[m], {m, 1, n}] + O[x]^(n+1), x] // Reverse;
row /@ Range[0, 8] // Flatten (* Jean-François Alcover, Sep 16 2019 *)
  • PARI
    tabl(nn) = {for (n=0, nn, polp = prod(k=1, n, x+prime(k)); forstep (k= n, 0, -1, print1(polcoeff(polp, k), ", ");); print(););} \\ Michel Marcus, Aug 10 2015

Formula

T(n, 1) = A007504(n) for n >= 1.
T(n, 2) = A024447(n) for n >= 2.

Extensions

Corrected and edited by M. F. Hasler, Aug 13 2015
a(20) in b-file corrected by Andrew Howroyd, Dec 31 2017

A309802 a(n) is the coefficient of x^n in the polynomial Product_{i=1..n+2} (prime(i)*x-1).

Original entry on oeis.org

1, 10, 101, 1358, 20581, 390238, 8130689, 201123530, 6166988769, 201097530280, 7754625545261, 329758834067168, 14671637258193181, 711027519310719868, 38706187989054920001, 2338431642812927422310, 145908145906128304198449, 9976861293427674211625032
Offset: 0

Views

Author

Alexey V. Bazhin, Aug 17 2019

Keywords

Links

Crossrefs

Cf. A000040, A002110, A024451, A070918, A309803, A309804, A033999, A007504, A024447, A024448, A024449, A054640, A005867, A238146, A260613.

Programs

  • Maple
    a:= n-> coeff(mul(ithprime(i)*x-1, i=1..n+2), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 18 2019

Formula

a(n) = [x^n] Product_{i=1..n+2} (prime(i)*x-1).
a(n) = abs(A070918(n+2,2)).
a(n) = abs(A238146(n+2,n)) for n>0.
a(n) = A260613(n+2,n).

A309803 a(n) is the coefficient of x^n in the polynomial Product_{i=1..n+3} (prime(i)*x-1).

Original entry on oeis.org

-1, -17, -288, -5102, -107315, -2429223, -64002818, -2057205252, -69940351581, -2788890538777, -122099137635118, -5580021752377242, -276932659619923555, -15388458479166668283, -946625238259888348698, -60082571176666116692888, -4171440414742758122621945
Offset: 0

Views

Author

Alexey V. Bazhin, Aug 17 2019

Keywords

Links

Crossrefs

Cf. A000040, A002110, A024451, A070918, A309802, A309804, A033999, A007504, A024447, A024448, A024449, A054640, A005867, A238146, A260613.

Programs

  • Maple
    a:= n-> coeff(mul(ithprime(i)*x-1, i=1..n+3), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 19 2019

Formula

a(n) = [x^n] Product_{i=1..n+3} (prime(i)*x-1).
a(n) = -abs(A070918(n+3,3)).
a(n) = -abs(A238146(n+3,n)) for n>0.
a(n) = -A260613(n+3,n).

A309804 a(n) is the coefficient of x^n in the polynomial Product_{i=1..n+4} (prime(i)*x-1).

Original entry on oeis.org

1, 28, 652, 16186, 414849, 11970750, 411154568, 14802996860, 617651235401, 28112591190218, 1330940558814492, 68134228016658366, 3888046744502816953, 244783216404832868510, 15878401438954693327808, 1123935467586630569656024, 83970858613393528568199649
Offset: 0

Views

Author

Alexey V. Bazhin, Aug 17 2019

Keywords

Crossrefs

Cf. A000040, A002110, A024451, A070918, A309802, A309803, A033999, A007504, A024447, A024448, A024449, A054640, A005867, A238146, A260613.

Programs

  • Maple
    a:= n-> coeff(mul(ithprime(i)*x-1, i=1..n+4), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 19 2019
  • Mathematica
    a[n_] := CoefficientList[Series[Product[Prime[i]*x - 1, {i, 1, n+4}], {x, 0, 25}], x] [[n+1]]; Array[a, 17, 0] (* Amiram Eldar, Aug 24 2019 *)
  • PARI
    a(n) = polcoef(prod(i=1, n+4, prime(i)*x-1), n); \\ Michel Marcus, Aug 25 2019

Formula

a(n) = [x^n] Product_{i=1..n+4} (prime(i)*x-1).
a(n) = abs(A070918(n+4,4)).
a(n) = abs(A238146(n+4,n)) for n>0.
a(n) = A260613(n+4,n).

A357176 a(n) is the least prime that is the n-th elementary symmetric function of the first k primes for some k.

Original entry on oeis.org

2, 31, 2101534937, 2927, 40361, 39075401846390482295581, 226026998201956974105518542793548663, 617651235401, 4325269278391458399931853204730438563, 12894795842691356733422939, 745410787149030809096434692201049325037186561467959704761393689387
Offset: 1

Views

Author

Robert Israel, Sep 21 2022

Keywords

Comments

a(n) is the first prime p such that (-1)^n*p is in the n-th column of A238146.

Examples

			a(4) = 2927 = 2*3*5*7 + 2*3*5*11 + 2*3*7*11 + 2*5*7*11 + 3*5*7*11 is the 4th symmetric function of the first 5 primes (2,3,5,7,11) and is prime.

Links

Crossrefs

Cf. A238146.

Programs

  • Maple
    N:= 20: V:= Vector(N):
S:= Vector(N): p:= 2: S[1]:= 2: V[1]:= 2: count:= 1:
while count < N do
  p:= nextprime(p);
  for k from N to 2 by -1 do
    S[k]:= S[k] + p*S[k-1];
    if V[k] = 0 and isprime(S[k]) then V[k]:= S[k]; count:= count+1; fi;
  od;
  S[1]:= S[1]+p;
od:
convert(V,list);
Showing 1-5 of 5 results.

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