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.

Previous Showing 11-20 of 29 results. Next

A321326 G.f. satisfies: A(x) = (1 - x) * Product_{k>0} A(x^(2*k)) / Product_{k>1} A(x^(2*k-1)).

Original entry on oeis.org

1, -1, -1, 2, -3, 2, 4, -6, -2, 5, 0, -3, 15, -13, -19, 27, -14, 4, 25, -33, -3, 7, 6, 15, 33, -48, -58, 69, -94, 119, 117, -245, 60, -3, -36, 170, 140, -237, -305, 300, -161, 248, 356, -554, 8, -263, 340, 469, -94, -25, -945, 102, -68, 1200, 694, -1786, 225, -974
Offset: 0

Views

Author

Seiichi Manyama, Nov 05 2018

Keywords

Links

Crossrefs

Cf. A067856, A321088, A321325, A321327.

Formula

G.f.: Product_{k>0} (1 - x^k)^A067856(k).
Product_{k>0} A(x^k) = Product_{k>=0} (1 - x^(2^k))^(2^k). (Cf. A321327.)

A354173 Product_{n>=1} (1 + x^(2*n))^(a(n)/(2*n)!) = cos(x).

Original entry on oeis.org

-1, -8, 104, -12032, 354944, 47546368, 6204652544, -6174957043712, 47215125069824, 159504062197792768, 51085990673656315904, 54592541528151763714048, 15510963121850795776016384, 14479308135716773591282352128, -7469518701197092988127633473536, -77646018400552596699424746364731392
Offset: 1

Views

Author

Ilya Gutkovskiy, May 18 2022

Keywords

Crossrefs

Cf. A067856, A170912, A170913, A354063, A354171, A354172, A354174, A354175, A354176.

Programs

  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[c[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; c[n_] := c[n] = {1, 0, -1, 0}[[Mod[n + 1, 4, 1]]]/n! - b[n, n - 1]; a[n_] := (2 n)! c[2 n]; Table[a[n], {n, 1, 16}]

Formula

E.g.f.: Sum_{k>=1} A067856(k) * log(cos(x^k)) / k (even powers only).

A354174 Product_{n>=1} (1 + x^(2*n))^(a(n)/(2*n)!) = cosh(x).

Original entry on oeis.org

1, 4, -104, 8128, -354944, -21642752, -6204652544, 4286437900288, -47215125069824, -78465506362130432, -51085990673656315904, -35027783166649488637952, -15510963121850795776016384, -7220202338641080038690127872, 7469518701197092988127633473536, 53919400066294168384184259715268608
Offset: 1

Views

Author

Ilya Gutkovskiy, May 18 2022

Keywords

Crossrefs

Cf. A067856, A353609, A354064, A354171, A354172, A354173, A354175, A354176.

Programs

  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[c[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; c[n_] := c[n] = Mod[n + 1, 2]/n! - b[n, n - 1]; a[n_] := (2 n)! c[2 n]; Table[a[n], {n, 1, 16}]

Formula

E.g.f.: Sum_{k>=1} A067856(k) * log(cosh(x^k)) / k (even powers only).

A332793 a(1) = 1; a(n) = n * Sum_{d|n, d < n} (-1)^(n/d) * a(d) / d.

Original entry on oeis.org

1, 2, -3, 8, -5, -6, -7, 32, 0, -10, -11, -24, -13, -14, 15, 128, -17, 0, -19, -40, 21, -22, -23, -96, 0, -26, 0, -56, -29, 30, -31, 512, 33, -34, 35, 0, -37, -38, 39, -160, -41, 42, -43, -88, 0, -46, -47, -384, 0, 0, 51, -104, -53, 0, 55, -224, 57, -58, -59, 120
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 24 2020

Keywords

Links

Crossrefs

Cf. A002129, A038838 (positions of 0's), A055615, A067856, A327268, A361987.
Partial sums give A361982.
Dirichlet inverse of A181983.

Programs

  • Mathematica
    a[1] = 1; a[n_] := n Sum[If[d < n, (-1)^(n/d) a[d]/d, 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 60}]
terms = 60; A[] = 0; Do[A[x] = x + Sum[(-1)^k k A[x^k], {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] // Rest
f[p_, e_] := If[p == 2, p^(2*e - 1), -p*Boole[e == 1]]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)

Formula

G.f. A(x) satisfies: A(x) = x + Sum_{k>=2} (-1)^k * k * A(x^k).
Dirichlet g.f.: 1 / (zeta(s-1) * (1 - 2^(2 - s))).
a(n) = Sum_{d|n} A327268(d).
Multiplicative with a(2^e) = 2^(2*e-1), and a(p^e) = -p if e=1 and 0 for e>1, for odd primes p. - Amiram Eldar, Dec 02 2020

A347030 a(n) = 1 + Sum_{k=2..n} (-1)^k * a(floor(n/k)).

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 0, 4, 4, 3, 2, 0, -1, -2, -1, 7, 6, 6, 5, 3, 4, 3, 2, -2, -2, -3, -3, -5, -6, -5, -6, 10, 11, 10, 11, 11, 10, 9, 10, 6, 5, 6, 5, 3, 3, 2, 1, -7, -7, -7, -6, -8, -9, -9, -8, -12, -11, -12, -13, -11, -12, -13, -13, 19, 20, 21, 20, 18, 19, 20, 19, 19, 18, 17, 17
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 11 2021

Keywords

Comments

Partial sums of A067856.

Links

Crossrefs

Cf. A002321, A025523, A067856, A309288, A347031.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + Sum[(-1)^k a[Floor[n/k]], {k, 2, n}]; Table[a[n], {n, 1, 75}]
nmax = 75; A[] = 0; Do[A[x] = (1/(1 - x)) (x + Sum[(-1)^k (1 - x^k) A[x^k], {k, 2, nmax}]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def A347030(n):
    if n <= 1:
        return n
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j&1)*(-1 if j&1 else 1)*A347030(k1)
        j, k1 = j2, n//j2
    return c+(n+1-j&1)*(-1 if j&1 else 1) # Chai Wah Wu, Apr 04 2023

Formula

G.f. A(x) satisfies: A(x) = (1/(1 - x)) * (x + Sum_{k>=2} (-1)^k * (1 - x^k) * A(x^k)).

A117997 Sum_{d|n} a(d) = n for n = 3^m (m >= 0) and for other n the sum is zero; i.e., the Möbius transform of [1, 0, 3, 0, 0, 0, 0, 0, 9, 0,...].

Original entry on oeis.org

1, -1, 2, 0, -1, -2, -1, 0, 6, 1, -1, 0, -1, 1, -2, 0, -1, -6, -1, 0, -2, 1, -1, 0, 0, 1, 18, 0, -1, 2, -1, 0, -2, 1, 1, 0, -1, 1, -2, 0, -1, 2, -1, 0, -6, 1, -1, 0, 0, 0, -2, 0, -1, -18, 1, 0, -2, 1, -1, 0, -1, 1, -6, 0, 1, 2, -1, 0, -2, -1, -1, 0, -1, 1, 0, 0, 1, 2, -1, 0, 54, 1, -1, 0, 1, 1, -2, 0, -1, 6, 1, 0, -2, 1, 1, 0, -1, 0, -6, 0, -1, 2, -1, 0, 2
Offset: 1

Views

Author

Paul D. Hanna, Apr 08 2006

Keywords

Comments

From Petros Hadjicostas, Jul 26 2020: (Start)
For p prime >= 2, Petrogradsky (2003) defined the multiplicative functions 1_p and mu_p in the following way:
1_p(n) = 1 when gcd(n,p) = 1 and 1_p(n) = 1 - p when gcd(n,p) = p;
mu_p(n) = mu(n) when gcd(n,p) = 1 and mu_p(n) = mu(m)*(p^s - p^(s-1)) when n = m*p^s with gcd(m,p) = 1 and s >= 1.
We have 1_2(n) = A062157(n), 1_3(n) = A061347(n), A067856(n) = mu_2(n), and a(n) = mu_3(n) for n >= 1.
Some of the results by other contributors here and in A067856 can be generalized:
(i) Rogel's (1897) formula for A067856 becomes Sum_{d | n} 1_p(d) * mu_p(n/d) = 0 for n > 1. Thus, 1_p is the Dirichlet inverse of mu_p.
(ii) R. J. Mathar's Dirichlet g.f. for mu_p becomes 1/(zeta(s) * (1 - p^(1-s))). The Dirichlet g.f. for 1_p is zeta(s) * (1 - p^(1-s)).
(iii) Benoit Cloitre's formula becomes 1 = Sum_{k=1..n} mu_p(k)*g_p(n/k), where g_p(x) = floor(x) - p*floor(x/p) = floor(x) mod p.
(iv) Paul D. Hanna's formula becomes Sum_{n >= 1} (mu_p(n)/n)*log((1 - x^(n*p))/(1 - x^n)) = x.
(v) The definition in the name of the sequence a(n) generalizes to Sum_{d | n} mu_p(d) = n, if n = p^s for s >= 0, and = 0, otherwise. Thus, mu_p(n) = Sum_{p^k | n, k >= 0} mu(n/p^k)*p^k. That is, (mu_p(n): n >= 1) is the Möbius transform of the sequence (b_p(n): n >= 1), where b_p(n) = p^k, if n = p^k for k >= 0, and b_p(n) = 0, otherwise.
(vi) We have the Lambert series Sum_{n >= 1} mu_p(n)*x^n/(1 - x^n) = Sum_{k >= 0} p^k*x^(p^k) = x + p*x^p + p^2*x^(p^2) + ..., which generalizes one of the formulas by Peter Bala in A067856.
(vii) By differentiating both sides of (iv) w.r.t. x and multiplying both sides by x, we get Sum_{n >= 1} mu_p(n)*(x^n + 2*x^(2*n) + ... + (p-1)*x^(n*(p-1)))/(1 + x^n + x^(2*n) + ... + x^(n*(p-1))) = x, which generalizes another one of Peter Bala's formulas in A067856. It can be thought as a "generalized Lambert series".
(viii) Dividing both sides of (vi) by x and integrating w.r.t. x from 0 to y, we get -Sum_{n >= 1} (mu_p(n)/n)*log(1 - y^n) = Sum_{k >= 0} y^(p^k) = y + y^p + y^(p^2) + y^(p^3) + ...
(ix) Obviously, f(n) = Sum_{d | n} 1_p(n/d)*g(d) if and only if g(n) = Sum_{d | n} mu_p(n/d)*f(d). (End)

Links

Crossrefs

Cf. A061347, A062157, A067856.

Programs

  • PARI
    {a(n)=if(n==1,1,-n*polcoeff(x+sum(k=1,n-1,a(k)/k*subst(log(1+x+x^2+x*O(x^n)),x,x^k+x*O(x^n))),n))}
  • PARI
    A117997(n) = sumdiv(n,d,moebius(n/d)*if((3^valuation(d,3))==d,d,0)); \\ Antti Karttunen, Jan 15 2025

Formula

G.f.: x = Sum_{n >= 1} (a(n)/n)*log(1 + x^n + x^(2*n)).
1 = Sum_{k=1..n} a(k)*g(n/k), where g(x) = floor(x) - 3*floor(x/3). [Benoit Cloitre, Nov 11 2010]
From Petros Hadjicostas, Jul 26 2020: (Start)
a(n) = Sum_{3^k | n, k >= 0} mu(n/3^k)*3^k.
Dirichlet g.f.: 1/(zeta(s)*(1 - 3^(1-s))).
The sequence is the Dirichlet inverse of A061347.
Sum_{n >= 1} a(n)*x^n/(1 - x^n) = x + 3*x^3 + 9*x^9 + 27*x^27 + 81*x^81 + ...
Sum_{n >= 1} a(n)*(x^n + 2*x^(2*n))/(1 + x^n + x^(2*n)) = x.
-Sum_{n >= 1} (a(n)/n)*log(1 - x^n) = x + x^3 + x^9 + x^27 + x^81 + ... (End)

Extensions

Offset changed to 1 by Petros Hadjicostas, Jul 26 2020

A259445 Multiplicative with a(n) = n if n is odd and a(2^s)=2.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 9, 10, 11, 6, 13, 14, 15, 2, 17, 18, 19, 10, 21, 22, 23, 6, 25, 26, 27, 14, 29, 30, 31, 2, 33, 34, 35, 18, 37, 38, 39, 10, 41, 42, 43, 22, 45, 46, 47, 6, 49, 50, 51, 26, 53, 54, 55, 14, 57, 58, 59, 30, 61, 62, 63, 2, 65, 66, 67, 34
Offset: 1

Views

Author

José María Grau Ribas, Jun 27 2015

Keywords

Comments

If n = 2^s*m with m odd and s > 0 then a(n) = 2*m.

Links

Crossrefs

Cf. A000265, A022998, A018819, A046897, A160467, A321558.

Programs

  • Maple
    A259445 := proc(n::integer)
    local a, pe, p,e ;
    a := 1 ;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        if p = 2 then
            a := 2*a ;
        else
            a := a*p^e  ;
        end if;
    end do:
    a;
end proc:
seq(A259445(n),n=1..80) ; # R. J. Mathar, Feb 21 2025
  • Mathematica
    G[n_] := If[Mod[n, 2] == 0, n/2^(FactorInteger[n][[1, 2]] - 1), n]; Table[G[n], {n, 1, 70}]
  • PARI
    a(n)=n>>max(valuation(n,2)-1,0) \\ Charles R Greathouse IV, Jun 28 2015

Formula

From Peter Bala, Feb 21 2019: (Start)
a(n) = n*gcd(n,2)/gcd(n,2^n).
a(2*n) = 2*A000265(2*n); a(2*n+1) = A000265(2*n+1).
O.g.f.: x*(1 + 4*x + x^2)/(1 - x^2)^2 - 2*( F(x^2) + F(x^4) + F(x^8) + ... ), where F(x) = x/(1 - x)^2.
O.g.f. for reciprocals: Sum_{n >= 1} (1/a(n))*x^n = (3/4)*L(x) - (1/4)*L(-x) + (1/4)*( L(x^2) + L(x^4) + L(x^8) + ... ), where L(x) = log(1/(1 - x)).
(End)
From Peter Bala, Mar 09 2019: (Start)
a(n) = (-1)^(n+1)*Sum_ {d divides n} (-1)^(d+n/d)*phi(d), where phi(n) = A000010(n) is the Euler totient function. Cf. the identity n = Sum_ {d divides n} phi(d). Cf. A046897 and A321558.
O.g.f.: Sum_{n >= 1} phi(n)*x^n/(1 + (-x)^n). (End)
From Amiram Eldar, Nov 28 2022: (Start)
Dirichlet g.f.: zeta(s-1)*(1 + 1/2^(s-1) - 2/(2^s-1)).
Sum_{k=1..n} a(k) ~ (5/12) * n^2. (End)
a(n) = n /A160467(n). - R. J. Mathar, Feb 21 2025

A321088 G.f. satisfies: A(x) = 1/(1 - x) * Product_{k>0} A(x^(2*k)) / Product_{k>1} A(x^(2*k-1)).

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 4, -1, 9, 3, 11, -4, 17, -2, 11, -24, 31, -3, 39, -35, 70, -14, 47, -107, 112, -27, 122, -163, 198, -90, 93, -409, 282, -108, 329, -487, 601, -160, 324, -1076, 835, -165, 907, -1298, 1478, -429, 565, -2973, 1745, -427, 1999, -3149, 3587, -528
Offset: 0

Views

Author

Seiichi Manyama, Nov 05 2018

Keywords

Links

Crossrefs

Convolution inverse of A321326.
Cf. A067856, A073709, A129374.

Programs

  • Mathematica
    b[n_] := If[n == 1, 1, Product[{p, e} = pe; If[2 == p, e--, If[e > 1, p = 0, p = -1]]; p^e, {pe, FactorInteger[n]}]];
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b];
a = etr[b];
a /@ Range[0, 100] (* Jean-François Alcover, Oct 01 2019 *)

Formula

Euler transform of A067856.
G.f.: Product_{k>0} 1/(1 - x^k)^A067856(k).
Product_{k>0} A(x^k) = Product_{k>=0} 1/(1 - x^(2^k))^(2^k). (Cf. A073709.)

A321559 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^3.

Original entry on oeis.org

1, -9, 28, -57, 126, -252, 344, -441, 757, -1134, 1332, -1596, 2198, -3096, 3528, -3513, 4914, -6813, 6860, -7182, 9632, -11988, 12168, -12348, 15751, -19782, 20440, -19608, 24390, -31752, 29792, -28089, 37296, -44226, 43344, -43149, 50654
Offset: 1

Views

Author

N. J. A. Sloane, Nov 23 2018

Keywords

Links

Crossrefs

Column k=3 of A322083.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[(-1)^(k+1)*k^3*x^k/(1 + x^k) : k in [1..2*m]]) )); // G. C. Greubel, Nov 28 2018
  • Mathematica
    a[n_] := DivisorSum[n, (-1)^(# + n/#)*#^3 &]; Array[a, 50] (* Amiram Eldar, Nov 27 2018 *)
  • PARI
    apply( A321559(n)=sumdiv(n, d, (-1)^(n\d-d)*d^3), [1..30]) \\ M. F. Hasler, Nov 26 2018
  • Sage
    s=(sum((-1)^(k+1)*k^3*x^k/(1 + x^k)  for k in (1..50))).series(x, 50); a = s.coefficients(x, sparse=False); a[1:] # G. C. Greubel, Nov 28 2018

Formula

G.f.: Sum_{k>=1} (-1)^(k+1)*k^3*x^k/(1 + x^k). - Ilya Gutkovskiy, Nov 27 2018
From Peter Bala, Jan 29 2022: (Start)
Multiplicative with a(2^k) = - 3*(2^(3*k+1) + 5)/7 for k >= 1 and a(p^k) = (p^(3*k+3) - 1)/(p^3 - 1) for odd prime p.
n^3 = (-1)^(n+1)*Sum_{d divides n} A067856(n/d)*a(d). (End)

A335062 a(n) = 1 - Sum_{d|n, d > 1} (-1)^d * a(n/d).

Original entry on oeis.org

1, 0, 2, 0, 2, -2, 2, 0, 4, -2, 2, 0, 2, -2, 6, 0, 2, -8, 2, 0, 6, -2, 2, 0, 4, -2, 8, 0, 2, -14, 2, 0, 6, -2, 6, 4, 2, -2, 6, 0, 2, -14, 2, 0, 16, -2, 2, 0, 4, -8, 6, 0, 2, -24, 6, 0, 6, -2, 2, 8, 2, -2, 16, 0, 6, -14, 2, 0, 6, -14, 2, 0, 2, -2, 16, 0, 6, -14, 2, 0, 16
Offset: 1

Views

Author

Ilya Gutkovskiy, May 21 2020

Keywords

Comments

Inverse Moebius transform of A308077.

Links

Crossrefs

Cf. A048298, A065091 (positions of 2's), A067824, A067856, A308077, A325144, A335283.

Programs

  • Mathematica
    a[n_] := a[n] = 1 - DivisorSum[n, (-1)^# a[n/#] &, # > 1 &]; Table[a[n], {n, 1, 81}]
  • PARI
    lista(nn) = {my(va = vector(nn)); for (n=1, nn, va[n] = 1 - sumdiv(n, d, if (d>1, (-1)^d*va[n/d]));); va;} \\ Michel Marcus, May 22 2020

Formula

G.f. A(x) satisfies: A(x) = x / (1 - x) - Sum_{k>=2} (-1)^k * A(x^k).
Previous Showing 11-20 of 29 results. Next

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

Content is available under The OEIS End-User License Agreement.