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.

A008667 Expansion of g.f.: 1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 10, 10, 13, 14, 17, 18, 22, 23, 28, 29, 34, 36, 42, 44, 50, 53, 60, 63, 71, 74, 83, 87, 96, 101, 111, 116, 127, 133, 145, 151, 164, 171, 185, 193, 207, 216, 232, 241, 258, 268, 286, 297, 316, 328, 348, 361, 382, 396, 419, 433, 457
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Also, Molien series for invariants of finite Coxeter group A_4. The Molien series for the finite Coxeter group of type A_k (k >= 1) has g.f. = 1/Product_{i=2..k+1} (1-x^i). Note that this is the root system A_k not the alternating group Alt_k. - N. J. A. Sloane, Jan 11 2016
Number of partitions into parts 2, 3, 4, and 5. - Joerg Arndt, Apr 29 2014

Examples

			a(4)=2 because f''''(x)/4!=2 at x=0 for f=1/((1-x^2)(1-x^3)(1-x^4)(1-x^5)).
G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 5*x^8 + 5*x^9 + 7*x^10 + 7*x^11 + ... .

References

  • J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.
  • L. Smith, Polynomial Invariants of Finite Groups, Peters, 1995, p. 199 (No. 32).

Links

Crossrefs

Molien series for finite Coxeter groups A_1 through A_12 are A059841, A103221, A266755, A008667, A037145, A001996, and A266776-A266781.
Cf. A005044, A001401 (partial sums).

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 65); Coefficients(R!( 1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)) )); // G. C. Greubel, Sep 08 2019
  • Maple
    seq(coeff(series(1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)), x, n+1), x, n), n = 0..65); # G. C. Greubel, Sep 08 2019
  • Mathematica
    SeriesCoefficient[1/((1-x^2)(1-x^3)(1-x^4)(1-x^5)),{x,0,#}]&/@Range[0,100] (* or *) a[k_]=SeriesCoefficient[1/((1-x^2)(1-x^3)(1-x^4) (1-x^5)),{x,0,k}] (* Peter Pein (petsie(AT)dordos.net), Sep 09 2006 *)
CoefficientList[Series[1/Times@@Table[(1-x^n),{n,2,5}],{x,0,70}],x] (* Harvey P. Dale, Feb 22 2018 *)
  • PARI
    {a(n) = if( n<-13, -a(-14 - n), polcoeff( prod( k=2, 5, 1 / (1 - x^k), 1 + x * O(x^n)), n))} /* Michael Somos, Oct 14 2006 */
  • Sage
    def A008667_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5))).list()
A008667_list(65) # G. C. Greubel, Sep 08 2019

Formula

Euler transform of length 5 sequence [ 0, 1, 1, 1, 1]. - Michael Somos, Sep 23 2006
a(-14 - n) = -a(n). - Michael Somos, Sep 23 2006
a(n) ~ 1/720*n^3. - Ralf Stephan, Apr 29 2014
a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-6) - 2*a(n-7) - a(n-8) + a(n-10) + a(n-11) + a(n-12) - a(n-14). - David Neil McGrath, Sep 13 2014
From R. J. Mathar, Jun 23 2021: (Start)
a(n)-a(n-2) = A008680(n).
a(n)-a(n-3) = A025802(n).
a(n)-a(n-4) = A025795(n).
a(n)-a(n-5) = A005044(n+3). (End)
a(n)= floor((n^3 + 21*n^2 + 156*n - 45*n*(n mod 2) + 720)/720 - [(n mod 10)=1]/5). - Hoang Xuan Thanh, Aug 20 2025

A029032 Expansion of 1/((1-x)*(1-x^3)*(1-x^4)*(1-x^5)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 17, 20, 23, 27, 31, 35, 40, 45, 51, 57, 63, 70, 78, 86, 94, 103, 113, 123, 134, 145, 157, 170, 183, 197, 212, 227, 243, 260, 278, 296, 315, 335, 356, 378, 400, 423, 448, 473, 499, 526, 554, 583, 613, 644, 676, 709, 743
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n) is the number of partitions of n into parts 1, 3, 4, and 5. - David Neil McGrath, Sep 13 2014

Links

Programs

  • Maple
    M := Matrix(13, (i,j)-> if (i=j-1) or (j=1 and member(i, [1, 3, 10, 12])) then 1 elif j=1 and member(i, [6, 7, 13]) then -1 else 0 fi); a := n -> (M^(n))[1,1]; seq (a(n), n=0..49); # Alois P. Heinz, Jul 25 2008
  • Mathematica
    CoefficientList[Series[1/((1-x)(1-x^3)(1-x^4)(1-x^5)),{x,0,50}],x] (* Harvey P. Dale, Jan 04 2012 *)

Formula

a(0)=1, a(1)=1, a(2)=1, a(3)=2, a(4)=3, a(5)=4, a(6)=5, a(7)=6, a(8)=8, a(9)=10, a(10)=12, a(11)=14, a(12)=17, a(n)=a(n-1)+a(n-3)-a(n-6)- a(n-7)+ a(n-10)+a(n-12)-a (n-13). - Harvey P. Dale, Jan 04 2012
From R. J. Mathar, Jun 23 2021: (Start)
a(n)-a(n-1) = A008680(n).
a(n)-a(n-3) = A025772(n).
a(n)-a(n-4) = A008672(n).
a(n)-a(n-5) = A025767(n). (End)
a(n) = 1 + floor((2*n^3+39*n^2+228*n)/720). - Hoang Xuan Thanh, May 29 2025

A384791 Numbers with a record number of ways in which they can be expressed as b^3 * c^4 * d^5, with b, c and d >= 1.

Original entry on oeis.org

1, 256, 4096, 32768, 262144, 1048576, 8388608, 16777216, 134217728, 268435456, 1073741824, 4294967296, 8589934592, 34359738368, 68719476736, 110075314176, 549755813888, 557256278016, 1761205026816, 4458050224128, 7044820107264, 8916100448256, 56358560858112, 71328803586048
Offset: 1

Views

Author

Amiram Eldar, Jun 10 2025

Keywords

Comments

The least term that is not a power of 2 is a(16) = 2^24 * 3^8.
Indices of records of the multiplicative function f(n) with f(p^e) = A008680(e).
All the terms are cubefull numbers since f(1) = 1 and f(n) = 0 if n is a noncubefull number.
The corresponding record values are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, ... (see the link for more values).
Every exponent must be the index of the first occurrence of A008680(e) in A008680. So possible exponents of prime factors of terms are 0, 8, 12, 15, 18, 20, ... - David A. Corneth, Jun 30 2025

Examples

			256 in the sequence as 256 = 1^3 * 4^4 * 1^5 = 2^3 * 1^4 * 2^5 so there are two ways to write 256 as b^3 * c^4 * d^5, with b, c and d >= 1 and no smaller positive integer can be written in at least two ways like that. - _David A. Corneth_, Jun 30 2025

Links

Crossrefs

Subsequence of A025487, A036966 and A181800.
Cf. A008680, A046055, A384789, A384790 (powerful analog).

Programs

  • Mathematica
    f[p_, e_] := Floor[(1+(-1)^e)*(-1)^Floor[e/2]/8 + (e^2 + 12*e + 90)/120]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; With[{lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]}, sm = -1; seq = {}; Do[s1 = s[lps[[i]]]; If[s1 > sm, sm = s1; AppendTo[seq, lps[[i]]]], {i, 1, Length[lps]}]; seq]

A384789 The number of ways in which the n-th cubefull number can be expressed as b^3 * c^4 * d^5, with b, c and d >= 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 4, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Jun 10 2025

Keywords

Comments

The positive values of the multiplicative function f(n) with f(p^e) = A008680(e). Or, equivalently, a(n) is the value of this function at A036966(n).

Examples

			a(12) = 2 since A036966(12) = 256 = 2^8 has 2 representations as b^3*c^4*d^5: 2^3 * 2^5 (b = d = 2, c = 1) and 4^4 (b = d = 1, c = 4).
a(38) = 3 since A036966(38) = 4096 = 2^12 has 3 representations as b^3*c^4*d^5: 2^3 * 2^4 * 2^5 (b = c = d = 2), 8^4 (b = d = 1, c = 8) and 16^3 (b = 16, c = d = 1).

Links

Crossrefs

Cf. A008680, A036966, A057523 (powerful analog), A384791.

Programs

  • Mathematica
    f[p_, e_] := Floor[(1+(-1)^e)*(-1)^Floor[e/2]/8 + (e^2 + 12*e + 90)/120]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 10000], # > 0 &]
  • PARI
    f(e) = floor((1+(-1)^e)*(-1)^floor(e/2)/8 + (e^2 + 12*e + 90)/120);
list(kmax) = {my(e); for(k = 1, kmax, e = factor(k)[, 2]; if(k == 1 || vecmin(e) > 2, print1(vecprod(apply(x -> f(x), e)), ", ")));}

Formula

a(n) >= 1.
Showing 1-4 of 4 results.

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

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