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 1-7 of 7 results.

A305882 -1 + Product_{n>=1} 1/(1 + a(n)*x^n) = g.f. of A000040 (prime numbers).

Original entry on oeis.org

-2, 1, 1, 4, 4, 13, 16, 44, 52, 112, 182, 411, 620, 1318, 2142, 5148, 7676, 15228, 27530, 58660, 98372, 207392, 364464, 763263, 1341508, 2773990, 4923220, 10470948, 18510902, 37546152, 69269976, 148419094, 258284232, 534761242, 981480012, 2004302204
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 13 2018

Keywords

Examples

			1/((1 - 2*x) * (1 + x^2) * (1 + x^3) * (1 + 4*x^4) * (1 + 4*x^5) * ... * (1 + a(n)*x^n) * ...) =  1 + 2*x + 3*x^2 + 5*x^3 + 7*x^4 + 11*x^5 + ... + A000040(k)*x^k + ...

Crossrefs

Cf. A000040, A030010, A030018, A145519, A147541, A147557, A305871, A305881.

Formula

Product_{n>=1} 1/(1 + a(n)*x^n) = 1 + Sum_{k>=1} prime(k)*x^k.
Product_{n>=1} (1 + a(n)*x^n) = Sum_{k>=0} A030018(k)*x^k.

A348128 Product_{n>=1} (1 + a(n)*x^n) = 1 + Sum_{n>=1} x^prime(n).

Original entry on oeis.org

0, 1, 1, 0, 0, 0, 1, 0, -1, -1, 2, 1, 0, -2, 0, 1, 3, -2, -1, 0, 4, 0, -1, -4, 6, 2, 2, -10, 4, 4, 13, -15, -7, -2, 30, -7, -7, -33, 42, 8, 16, -70, 27, 22, 95, -116, -21, -39, 223, -61, -48, -261, 326, 51, 129, -581, 242, 109, 752, -932, -105, -330, 1806, -612, -240, -2140, 2750, 227, 1245, -4865
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 01 2021

Keywords

Links

Crossrefs

Cf. A000040, A002121, A010051, A030010, A147557, A305871, A305882, A308298, A328777, A348127.

A328777 Product_{n>=1} (1 + x^n)^a(n) = 1 + Sum_{n>=1} x^prime(n).

Original entry on oeis.org

0, 1, 1, 0, 0, 0, 1, 0, -1, -1, 2, 1, 0, -2, 0, 1, 3, -3, -1, -1, 4, -1, -1, -4, 6, 2, 2, -13, 4, 4, 13, -15, -5, -5, 30, -11, -7, -34, 42, 7, 16, -78, 27, 18, 95, -117, -21, -49, 223, -76, -40, -262, 326, 48, 135, -642, 242, 103, 752, -938, -105, -408, 1826, -732, -240
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 27 2019

Keywords

Comments

Inverse weigh transform of A010051.

Crossrefs

Cf. A000040, A000586, A010051, A305871, A308298.

Programs

  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; a[n_] := a[n] = Boole[PrimeQ[n]] - b[n, n - 1]; Array[a, 65]

A348127 Product_{n>=1} 1 / (1 - a(n)*x^n) = 1 + Sum_{n>=1} x^prime(n).

Original entry on oeis.org

0, 1, 1, -1, 0, -1, 1, -1, -1, -1, 2, 0, 0, -3, 0, 0, 3, -3, -1, -1, 4, -4, -1, -5, 6, 2, 2, -17, 4, 4, 13, -16, -7, -11, 30, -14, -7, -34, 42, 7, 16, -80, 27, 6, 95, -117, -21, -60, 223, -97, -48, -265, 326, 53, 129, -800, 242, 93, 752, -948, -105, -499, 1806, -853, -240, -2189, 2750, 124
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 01 2021

Keywords

Links

Crossrefs

Cf. A000040, A002121, A010051, A030010, A147557, A305871, A305882, A308298, A328777, A348128.

Programs

  • Maple
    N:= 20: # for a(1)..a(N)
P:= 1: a:= Vector(N):
for n from 1 to N do
  c:= coeff(P,x,n);
  if isprime(n) then a[n]:= 1-c  else a[n]:= -c fi;
  P:= series(P/(1-a[n]*x^n),x,N+1);
od:
convert(a,list); # Robert Israel, Mar 01 2022

A353160 Product_{n>=1} (1 + x^n)^a(n) = 1 + Sum_{n>=1} prime(n+1) * x^n.

Original entry on oeis.org

3, 2, 0, 4, -4, 6, -8, 11, 0, -18, 40, -67, 88, -78, -4, 205, -524, 926, -1234, 1060, 140, -2998, 7900, -14132, 18858, -16280, -2710, 48783, -127826, 228784, -306494, 263582, 55144, -829538, 2160120, -3864533, 5172792, -4406277, -1118324, 14531101, -37606792
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 28 2022

Keywords

Comments

Inverse weigh transform of odd primes.

Crossrefs

Cf. A006005, A065091, A305871, A352609, A353161.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= proc(n) option remember; ithprime(n+1)-b(n, n-1) end:
seq(a(n), n=1..50);  # Alois P. Heinz, Apr 28 2022
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; a[n_] := a[n] = Prime[n + 1] - b[n, n - 1]; Table[a[n], {n, 1, 41}]

A353161 Product_{n>=1} (1 + x^n)^a(n) = 1 + x + Sum_{n>=2} prime(n-1) * x^n.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, -1, 1, -3, -1, 4, 1, 5, 2, -4, -4, -9, 0, -3, 14, 19, 4, 6, -38, -27, -17, 5, 59, 50, 103, -49, -100, -142, -222, 83, 138, 468, 362, 0, -313, -1215, -599, -526, 961, 2572, 1837, 1673, -2858, -4516, -6182, -3880, 5981, 9282, 18218, 7414, -8554, -24446
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 28 2022

Keywords

Comments

Inverse weigh transform of {1, primes}.

Crossrefs

Cf. A008578, A030011, A305871, A353160.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= proc(n) option remember; `if`(n=1, 1, ithprime(n-1))-b(n, n-1) end:
seq(a(n), n=1..60);  # Alois P. Heinz, Apr 28 2022
  • Mathematica
    p[n_] := If[n == 1, 1, Prime[n - 1]]; b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; a[n_] := a[n] = p[n] - b[n, n - 1]; Table[a[n], {n, 1, 57}]

A380614 Product_{n>=1} (1 + x^n)^a(n) = Sum_{n>=0} prime(n)# * x^n.

Original entry on oeis.org

2, 5, 20, 155, 1860, 24970, 444060, 8583935, 202071920, 5992773714, 186947632200, 7001535728810, 288868991951760, 12455290280871150, 587972068547997856, 31327583556949986095, 1856116108295418943020, 113366872636395467452840, 7619343577986975410930880, 541957669076266404650853414
Offset: 1

Views

Author

Ilya Gutkovskiy, Jan 28 2025

Keywords

Comments

Inverse Weigh transform of primorial numbers.

Links

  • Eric Weisstein's World of Mathematics, Primorial.

Crossrefs

Cf. A002110, A168246, A305871, A380498, A380613.

Programs

  • Maple
    p:= proc(n) option remember; `if`(n<1, 1, p(n-1)*ithprime(n)) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= proc(n) option remember; p(n)-b(n, n-1) end:
seq(a(n), n=1..20);  # Alois P. Heinz, Jan 28 2025
  • Mathematica
    primorial[n_] := Product[Prime[j], {j, 1, n}]; b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; a[n_] := a[n] = primorial[n] - b[n, n - 1]; Array[a, 20]
Showing 1-7 of 7 results.

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