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.

A002122 a(n) = Sum_{t=0..n} g(t)*g(n-t) where g(t) = A002121(t).

Original entry on oeis.org

1, 0, -2, 2, 3, -4, -1, 8, -1, -10, 9, 16, -18, -12, 42, 4, -58, 40, 82, -88, -54, 188, 18, -248, 151, 354, -338, -260, 760, 120, -1031, 574, 1460, -1324, -1076, 2948, 542, -3962, 2075, 5644, -4868, -4290, 11035, 2418, -14900, 7346, 21300, -17652, -16323, 40442, 9768, -54476, 25675, 78290, -62456
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Arises in studying the Goldbach conjecture.
The last negative term appears to be a(485). - T. D. Noe, Dec 05 2006

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002121.

Programs

  • Haskell
    a002122 n = a002122_list !! n
a002122_list = uncurry conv $ splitAt 1 a002121_list where
   conv xs (z:zs) = sum (zipWith (*) xs $ reverse xs) : conv (z:xs) zs
-- Reinhard Zumkeller, Mar 21 2014

Formula

G.f.: 1/(1+Sum_{k>0} (-x)^prime(k))^2.

Extensions

Edited by Vladeta Jovovic, Mar 29 2003
Entry revised by N. J. A. Sloane, Dec 04 2006

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.

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

A329098 Expansion of 1 / (1 + Sum_{p prime, k>=1} x^(p^k)).

Original entry on oeis.org

1, 0, -1, -1, 0, 1, 2, 0, -3, -3, 2, 5, 4, -4, -10, -5, 10, 16, 5, -20, -27, 0, 41, 38, -14, -73, -55, 46, 134, 63, -118, -219, -55, 252, 356, -11, -510, -527, 198, 951, 734, -644, -1702, -867, 1579, 2864, 764, -3415, -4609, 84, 6808, 6897, -2526, -12745, -9539, 8383
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 04 2019

Keywords

Crossrefs

Cf. A002121, A246655, A280195.

Programs

  • Mathematica
    nmax = 55; CoefficientList[Series[1/(1 + Sum[Boole[PrimePowerQ[k]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -Sum[Boole[PrimePowerQ[k]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 55}]

Formula

G.f.: 1 / (1 + Sum_{k>=1} x^A246655(k)).

A329099 Expansion of 1 / (1 + Sum_{k>=1} mu(k)^2 * x^k).

Original entry on oeis.org

1, -1, 0, 0, 1, -2, 1, 0, 2, -4, 2, 0, 4, -10, 7, 0, 7, -23, 22, -6, 14, -51, 59, -24, 31, -113, 152, -80, 66, -244, 383, -253, 166, -521, 930, -746, 460, -1133, 2219, -2082, 1314, -2494, 5208, -5607, 3788, -5622, 12037, -14608, 10830, -13145, 27618, -37089, 30350, -31914, 63248, -92290
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 04 2019

Keywords

Crossrefs

Cf. A002121, A005117, A008683, A280194.

Programs

  • Mathematica
    nmax = 55; CoefficientList[Series[1/(1 + Sum[MoebiusMu[k]^2 x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -Sum[Boole[SquareFreeQ[k]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 55}]

Formula

G.f.: 1 / (1 + Sum_{k>=1} x^A005117(k)).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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