A129353 -id:A129353 - cp's OEIS frontend

A129360 A054525 * A115361.

1, 0, 1, -1, 0, 1, 0, 0, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Apr 10 2007

Row sums = A209229 (1, 1, 0, 1, 0, 0, 0, 1, ...).

A129353 = the inverse Möbius transform of A115361.

			First few rows of the triangle are:
   1;
   0,  1;
  -1,  0,  1;
   0,  0,  0,  1;
  -1,  0,  0,  0,  1;
   0, -1,  0,  0,  0,  1;
  -1,  0,  0,  0,  0,  0,  1;
   0,  0,  0,  0,  0,  0,  0,  1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Column 1 is A087003 (Moebius transform of A209229).
Row sums are A209229.
Cf. A054525, A115361, A129353, A001511, A103994, A129501.

tabl(nn) = {Tm = matrix(nn, nn, n, k, if (! (n % k), moebius(n/k), 0)); Tr = matrix(nn, nn, n, k, n--; k--; if ((n==k), 1, if (n==2*k+1, -1, 0))); Ti = Tr^(-1); Tp = Tm*Ti; for (n=1, nn, for (k=1, n, print1(Tp[n, k], ", ");); print(););} \\ Michel Marcus, Mar 28 2015

T(n, k)={ if(n%k, 0, sumdiv(n/k, d, my(e=valuation(d, 2)); if(d==1<Andrew Howroyd, Aug 03 2018

Moebius transform of A115361.

T(n,k) = A087003(n/k) for k | n, T(n,k) = 0 otherwise. - Andrew Howroyd, Aug 03 2018

More terms from Michel Marcus, Mar 28 2015

Offset changed by Andrew Howroyd, Aug 03 2018

A129628 Inverse Moebius transform of A001511.

Original entry on oeis.org

1, 3, 2, 6, 2, 6, 2, 10, 3, 6, 2, 12, 2, 6, 4, 15, 2, 9, 2, 12, 4, 6, 2, 20, 3, 6, 4, 12, 2, 12, 2, 21, 4, 6, 4, 18, 2, 6, 4, 20, 2, 12, 2, 12, 6, 6, 2, 30, 3, 9, 4, 12, 2, 12, 4, 20, 4, 6, 2, 24, 2, 6, 6, 28, 4, 12, 2, 12, 4, 12, 2, 30, 2, 6, 6, 12, 4, 12, 2, 30, 5, 6, 2, 24, 4, 6, 4, 20

Offset: 1

Ralf Stephan, May 31 2007

Dirichlet convolution of A000005 with A209229. - Ridouane Oudra, Jul 25 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Andrew Howroyd)

Row sums of A129353.
Cf. A000005, A001511, A001620, A129628.
Cf. A007814, A099777, A115364, A001227, A209229.

seq(add(padic[ordp](2*d, 2), d in numtheory[divisors](n)), n=1..100); # Ridouane Oudra, Sep 30 2024

f[p_, e_] := If[p==2, (e+1)*(e+2)/2, e+1]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 30 2020 *)

a(n)={sumdiv(n, d, 1 + valuation(d, 2))} \\ Andrew Howroyd, Aug 04 2018

a(2n) = a(n) + A000005(2n), a(2n+1) = A000005(2n+1).
Dirichlet g.f.: zeta(s)^2 * 2^s/(2^s-1). - Ralf Stephan, Jun 17 2007, corrected by Vaclav Kotesovec, Feb 02 2019
a(n) = Sum_{d|n} A001511(d). - Andrew Howroyd, Aug 04 2018
Sum_{k=1..n} a(k) ~ 2*n * (2*gamma - 1 + log(n/2)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 02 2019
Multiplicative with a(2^e) = (e+1)*(e+2)/2, and a(p^e) = e+1 for p > 2. - Amiram Eldar, Sep 30 2020
From Ridouane Oudra, Sep 30 2024: (Start)
a(n) = Sum_{i=0..A007814(n)} tau(n/2^i).
a(n) = Sum_{d|2*n} A007814(d).
a(n) = (1/2)*A001511(n)*A099777(n).
a(n) = (1/2)*(A001511(n) + 1)*A000005(n).
a(n) = A115364(n)*A001227(n). (End)

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A129360 A054525 * A115361.

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A129628 Inverse Moebius transform of A001511.

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