A115364 -id:A115364 - cp's OEIS frontend

A129628 Inverse Moebius transform of A001511.

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)

A094290 a(n) = prime(A001511(n)), where A001511 is one more than the 2-adic valuation of n.

Original entry on oeis.org

2, 3, 2, 5, 2, 3, 2, 7, 2, 3, 2, 5, 2, 3, 2, 11, 2, 3, 2, 5, 2, 3, 2, 7, 2, 3, 2, 5, 2, 3, 2, 13, 2, 3, 2, 5, 2, 3, 2, 7, 2, 3, 2, 5, 2, 3, 2, 11, 2, 3, 2, 5, 2, 3, 2, 7, 2, 3, 2, 5, 2, 3, 2, 17, 2, 3, 2, 5, 2, 3, 2, 7, 2, 3, 2, 5, 2, 3, 2, 11, 2, 3, 2, 5, 2, 3, 2, 7, 2, 3, 2, 5, 2, 3, 2, 13, 2, 3, 2, 5, 2, 3

Offset: 1

Amarnath Murthy, Apr 28 2004

nonn

Originally defined as: a(1) = 2 = prime(1). Then the first occurrence of prime(n) followed by all previous terms. i.e. If the index of first occurrence of prime(n) is k then the next k-1 terms are defined as a(k+r) = a(r), r = 1 to k-1. and a(2k) = prime(n+1) and so on.
Index of the first occurrence of prime(n)= 2^(n-1). Subsidiary sequences: If prime(n) is replaced by f(n) a large number of sequences can be obtained choosing f(n) = composite(n), f(n) = n^2,f(n) = n^r, r =3,4,5,..., f(n) = tau(n), f(n) = sigma(n), f(n) = n!, f(n) = Fibonacci(n), f(n) = T(n), triangular number, f(n) = n-th Bell, etc. each giving a distinct fascinating music.
The lexicographically earliest sequence such that no product of consecutive terms is a perfect square. - Joshua Zucker, Apr 30 2011

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000040, A001511.

Cf. also A115364.

Array[Prime[IntegerExponent[#, 2] + 1] &, 102] (* Michael De Vlieger, Nov 02 2018 *)

A094290(n) = prime(1+valuation(n,2)); \\ Antti Karttunen, Nov 02 2018

from sympy import prime
def A094290(n): return prime((~n & n-1).bit_length()+1) # Chai Wah Wu, Jul 02 2022

a(n) = A000040(A001511(n)). - Omar E. Pol, Sep 13 2013

Replaced the name with a formula given by Omar E. Pol, which is equivalent to the original definition. - Antti Karttunen, Nov 02 2018

A316631 Expansion of A(x) = x(1+3x^2+x^3+3*x^4+x^6)/(1-x^4)^2.

Original entry on oeis.org

0, 1, 0, 3, 1, 5, 0, 7, 2, 9, 0, 11, 3, 13, 0, 15, 4, 17, 0, 19, 5, 21, 0, 23, 6, 25, 0, 27, 7, 29, 0, 31, 8, 33, 0, 35, 9, 37, 0, 39, 10, 41, 0, 43, 11, 45, 0, 47, 12, 49, 0, 51, 13, 53, 0, 55, 14, 57, 0, 59, 15, 61, 0, 63, 16, 65, 0, 67, 17, 69, 0, 71, 18, 73, 0, 75, 19, 77, 0, 79, 20

Offset: 0

Werner Schulte, Jul 09 2018

			a(22) = 0 since 22 mod 4 = 2; a(23) = 23 for 23 mod 2 = 1; a(24) = 6 because 24 mod 4 = 0 and 24/4 = 6.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature(0,0,0,2,0,0,0,-1).

Cf. A000027, A000203, A104117, A115364.

a:=[0,1,0,3,1,5,0,7];; for n in [9..85] do a[n]:=2*a[n-4]-a[n-8]; od; a; # Muniru A Asiru, Jul 20 2018

seq(coeff(series(x*(1+3*x^2+x^3+3*x^4+x^6)/(1-x^4)^2, x,n+1),x,n),n=0..80); # Muniru A Asiru, Jul 20 2018

CoefficientList[Series[x (1 + 3 x^2 + x^3 + 3 x^4 + x^6)/(1 - x^4)^2, {x, 0, 80}], x] (* Michael De Vlieger, Jul 20 2018 *)
LinearRecurrence[{0, 0, 0, 2, 0, 0, 0, -1}, {0, 1, 0, 3, 1, 5, 0, 7}, 81] (* Robert G. Wilson v, Jul 21 2018 *)

concat(0, Vec((x*(1+3*x^2+x^3+3*x^4+x^6)/(1-x^4)^2) + O(x^80))) \\ Felix Fröhlich, Jul 09 2018

{my(N=79); concat([0], dirdiv(vector(N,n,n), vector(N, n, my(k=valuation(n, 2)); if(n==2^k, k+1, 0))))} \\ Andrew Howroyd, Jul 09 2018

a(n) = n/4 if n mod 4 = 0, and a(n) = 0 if n mod 4 = 2, and a(n) = n if n mod 2 = 1.
Linear recurrence: a(n) = 2*a(n-4) - a(n-8) for n > 7.
a(n) for n > 0 is multiplicative with a(2^e) = 1 - e if e < 2 and a(2^e) = 2^(e-2) if e > 1 otherwise a(p^e) = p^e for prime p > 2 and e >= 0.
Dirichlet g.f.: Sum_{n>0} a(n)/n^s = (1-1/2^s)^2 * zeta(s-1).
Dirichlet inverse b(n) for n > 0 is multiplicative with b(2^e) = 1 - e and for prime p > 2: b(p) = -p and b(p^e) = 0 if e > 1.
Dirichlet convolution with A104117(n) yields A000027(n).
Dirichlet convolution with A115364(n) yields A000203(n).
Sum_{k=1..n} a(k) ~ (9/32) * n^2. - Amiram Eldar, Nov 20 2022

A373438 Expansion of Sum_{k>=1} k * x^(3^(k-1)) / (1 - x^(3^(k-1))).

Original entry on oeis.org

1, 1, 3, 1, 1, 3, 1, 1, 6, 1, 1, 3, 1, 1, 3, 1, 1, 6, 1, 1, 3, 1, 1, 3, 1, 1, 10, 1, 1, 3, 1, 1, 3, 1, 1, 6, 1, 1, 3, 1, 1, 3, 1, 1, 6, 1, 1, 3, 1, 1, 3, 1, 1, 10, 1, 1, 3, 1, 1, 3, 1, 1, 6, 1, 1, 3, 1, 1, 3, 1, 1, 6, 1, 1, 3, 1, 1, 3, 1, 1, 15, 1, 1, 3, 1, 1, 3, 1, 1, 6, 1, 1, 3, 1, 1, 3, 1, 1, 6, 1, 1, 3, 1, 1, 3

Offset: 1

Ilya Gutkovskiy, Jun 05 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000217, A051064, A080278, A115364.

nmax = 105; CoefficientList[Series[Sum[k x^(3^(k - 1))/(1 - x^(3^(k - 1))), {k, 1, Floor[Log[3, nmax]] + 1}], {x, 0, nmax}], x] // Rest
Table[Binomial[IntegerExponent[3 n, 3] + 1, 2], {n, 1, 105}]

a(n) = {my(e = valuation(n, 3)); (e+1)*(e+2)/2;} \\ Amiram Eldar, Jun 27 2024

a(n) = A000217(A051064(n)).
From Vaclav Kotesovec, Jun 25 2024: (Start)
Dirichlet g.f.: zeta(s) * (3^s/(3^s-1))^2.
Sum_{k=1..n} a(k) ~ 9*n/4 - log(n)*(log(n) + 2*log(6*Pi))/(4*log(3)^2). (End)
Multiplicative with a(p^e) = (e+1)*(e+2)/2 if p = 3 and 1 if p != 3. - Amiram Eldar, Jun 27 2024

A115363 ((1,x)-(x,x^2))^(-2) (using Riordan array notation).

Original entry on oeis.org

1, 2, 1, 0, 0, 1, 3, 2, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 4, 3, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 3, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Paul Barry, Jan 21 2006

Square of number triangle A115361. Row sums are A115364.

			Triangle begins
1,
2, 1,
0, 0, 1,
3, 2, 0, 1,
0, 0, 0, 0, 1,
0, 0, 2, 0, 0, 1,
0, 0, 0, 0, 0, 0, 1,
4, 3, 0, 2, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 2, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
0, 0, 3, 0, 0, 2, 0, 0, 0, 0, 0, 1,

A349693 Dirichlet convolution of the ruler function (A001511) with itself.

Original entry on oeis.org

1, 4, 2, 10, 2, 8, 2, 20, 3, 8, 2, 20, 2, 8, 4, 35, 2, 12, 2, 20, 4, 8, 2, 40, 3, 8, 4, 20, 2, 16, 2, 56, 4, 8, 4, 30, 2, 8, 4, 40, 2, 16, 2, 20, 6, 8, 2, 70, 3, 12, 4, 20, 2, 16, 4, 40, 4, 8, 2, 40, 2, 8, 6, 84, 4, 16, 2, 20, 4, 16, 2, 60, 2, 8, 6, 20, 4, 16, 2, 70

Offset: 1

Ilya Gutkovskiy, Nov 25 2021

Dirichlet convolution of A000005 with A104117. - Ridouane Oudra, Jul 23 2025

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000217, A001511, A115364, A129628.

Cf. A000005, A104117, A007814, A085058, A090739, A099777, A372784, A001227, A000292,

a:= n-> (f-> add(f(d)*f(n/d), d=numtheory[divisors](n)))(k-> padic[ordp](2*k, 2)):
seq(a(n), n=1..80);  # Alois P. Heinz, Nov 25 2021

Table[Sum[IntegerExponent[2 d, 2] IntegerExponent[2 n/d, 2], {d, Divisors[n]}], {n, 1, 80}]
f[p_, e_] := If[p == 2, Binomial[e + 3, 3], e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 80] (* Amiram Eldar, Nov 25 2021 *)

A001511(n) = (1+valuation(n,2));
A349693(n) = sumdiv(n,d,A001511(n/d)*A001511(d)); \\ Antti Karttunen, Nov 25 2021

from sympy import divisor_count
def A349693(n): return divisor_count(n)*(m:=(n&-n).bit_length()+1)*(m+1)//6 # Chai Wah Wu, Jul 13 2022

Dirichlet g.f.: zeta(s)^2 * 4^s / (2^s-1)^2.
a(n) = Sum_{d|n} A001511(d) * A001511(n/d).
a(n) = Sum_{d|n} A000217(A001511(d)).
Multiplicative with a(p^e) = binomial(e+3,3) if p = 2 and e+1 otherwise. - Amiram Eldar, Nov 25 2021
Sum_{k=1..n} a(k) ~ 4*n*(log(n) - 1 + 2*gamma - 2*log(2)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Nov 26 2021
From Ridouane Oudra, Jul 23 2025: (Start)
a(n) = Sum_{i=0..A007814(n)} (i+1)*tau(n/2^i).
a(n) = Sum_{d|n} A115364(d).
a(n) = (1/6)*A090739(n)*A085058(n-1)*A000005(n).
a(n) = (1/6)*A001511(n)*A090739(n)*A099777(n).
a(n) = (1/3)*A115364(n)*A372784(n).
a(n) = A001227(n)*A000292(A001511(n)).
a(2*n+1) = tau(2*n+1).
a(2^k*(2*n+1)) = binomial(k+3, 3)*tau(2*n+1), for k, n >= 0. (End)

cp's OEIS Frontend

A129628 Inverse Moebius transform of A001511.

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A094290 a(n) = prime(A001511(n)), where A001511 is one more than the 2-adic valuation of n.

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A316631 Expansion of A(x) = x*(1+3*x^2+x^3+3*x^4+x^6)/(1-x^4)^2.

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A373438 Expansion of Sum_{k>=1} k * x^(3^(k-1)) / (1 - x^(3^(k-1))).

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A115363 ((1,x)-(x,x^2))^(-2) (using Riordan array notation).

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A349693 Dirichlet convolution of the ruler function (A001511) with itself.

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Maple

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A316631 Expansion of A(x) = x(1+3x^2+x^3+3*x^4+x^6)/(1-x^4)^2.