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 1-4 of 4 results.

A002098 G.f.: 1/Product_{k>=1} (1-prime(k)*x^prime(k)).

Original entry on oeis.org

1, 0, 2, 3, 4, 11, 17, 29, 49, 85, 144, 226, 404, 603, 1025, 1679, 2558, 4201, 6677, 10190, 16599, 25681, 39643, 61830, 96771, 147114, 228338, 352725, 533291, 818624, 1263259, 1885918, 2900270, 4396577, 6595481, 10040029, 15166064, 22642064
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n) is sum of all numbers k for which A001414(k), the sum of prime factors with repetition, equals n. See Havermann's link. - J. M. Bergot, Jun 14 2013

References

  • S.M. Kerawala, On a Pair of Arithmetic Functions Analogous to Chawla's Pair, J. Natural Sciences and Mathematics, 9 (1969), circa p. 103.
  • 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. A002099, A006906.
Row sums of A064364, A116864.

Programs

  • Maple
    b:= proc(n,i) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i=0 then 0
    else b(n, i-1) +b(n-ithprime(i), i) *ithprime(i)
      fi
    end:
a:= n-> b(n, numtheory[pi](n)):
seq(a(n), n=0..40);  # Alois P. Heinz, Nov 20 2010
  • Mathematica
    b[n_, i_] := b[n, i] = Which[n<0, 0, n==0, 1, i==0, 0, True, b[n, i-1] + b[n - Prime[i], i]*Prime[i]]; a[n_] := b[n, PrimePi[n]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 19 2016, after Alois P. Heinz *)
With[{nn=40},CoefficientList[Series[1/Product[1-Prime[k]x^Prime[k],{k,nn}],{x,0,nn}],x]] (* Harvey P. Dale, Jun 20 2021 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-isprime(k)*k*x^k)) \\ Seiichi Manyama, Feb 27 2022

Extensions

Better description and more terms from Vladeta Jovovic, May 09 2003

A298160 G.f.: Product_{k>=1} 1/(1+prime(k)*x^prime(k)).

Original entry on oeis.org

1, 0, -2, -3, 4, 1, 1, -9, 13, -9, 20, -38, 76, -75, 65, -323, 378, -197, 805, -1394, 1635, -2513, 3175, -5442, 11135, -12570, 12526, -33357, 51563, -46460, 93551, -155750, 186650, -313241, 421641, -620393, 1131820, -1321220, 1663951, -3559915, 5011036, -5207116
Offset: 0

Views

Author

Seiichi Manyama, Jan 14 2018

Keywords

Links

Crossrefs

Cf. A002099, A048165, A298159.

Programs

  • Mathematica
    CoefficientList[Series[Product[1/(1+Prime[k]x^Prime[k]),{k,50}],{x,0,50}],x] (* Harvey P. Dale, Mar 24 2020 *)

A305881 Expansion of Product_{k>=1} 1/(1 + prime(k)*x^k).

Original entry on oeis.org

1, -2, 1, -7, 16, -28, 62, -118, 303, -630, 1152, -2426, 5315, -10718, 20482, -43449, 91111, -179254, 358910, -727829, 1484601, -2995681, 5924606, -11935441, 24382120, -48702245, 96682698, -195063604, 392983826, -784903199, 1569490057, -3146479152, 6317124649, -12652202092
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 13 2018

Keywords

Comments

Convolution inverse of A147655.

Links

Crossrefs

Cf. A000040, A002099, A061151, A145519, A147655, A298160, A304791, A305882.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, b(n-i, i-1)*ithprime(i))))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      -add(b(n-i$2)*a(i$2), i=0..n-1))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 13 2018
  • Mathematica
    nmax = 33; CoefficientList[Series[Product[1/(1 + Prime[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 33; CoefficientList[Series[Exp[Sum[Sum[(-1)^k Prime[j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (-Prime[d])^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 33}]

Formula

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^k*prime(j)^k*x^(j*k)/k).

A319113 Expansion of e.g.f. Product_{k>=1} (1 + x^prime(k)/prime(k)).

Original entry on oeis.org

1, 0, 1, 2, 0, 44, 0, 1224, 2688, 25920, 293760, 3628800, 25090560, 762048000, 3887170560, 62749209600, 1233908121600, 22616539545600, 321930878976000, 10717413809356800, 108951843667968000, 1982497256570880000, 50138292140310528000, 1408088823809310720000, 25175914255793258496000
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 10 2018

Keywords

Crossrefs

Cf. A002099, A007838, A319112.

Programs

  • Maple
    seq(n!*coeff(series(mul((1+x^ithprime(k)/ithprime(k)),k=1..100),x=0,25),x,n),n=0..24); # Paolo P. Lava, Jan 09 2019
  • Mathematica
    nmax = 24; CoefficientList[Series[Product[(1 + x^Prime[k]/Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 24; CoefficientList[Series[Exp[Sum[Sum[Boole[PrimeQ[d]] (-d)^(1 - k/d), {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[Sum[Boole[PrimeQ[d]] (-d)^(1 - k/d), {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 24}]
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, 1+isprime(k)*x^k/k))) \\ Seiichi Manyama, Feb 27 2022

Formula

E.g.f.: exp(Sum_{k>=1} ( Sum_{p|k, p prime} (-p)^(1-k/p) ) * x^k/k).
Showing 1-4 of 4 results.

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

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