A129527 -id:A129527 - cp's OEIS frontend

A382335 Expansion of ( 1 + 4 * Sum_{k>=0} x^(2^k)/(1 - x^(2^k))^2 )^(1/2).

1, 2, 4, -2, 10, -2, -20, 82, -108, -114, 1052, -2702, 2054, 11394, -52636, 99534, 32938, -831698, 2649676, -3119694, -8779530, 54334130, -125649628, 31877726, 849214460, -3274210670, 5129552132, 7097067566, -65583106070, 180299051838, -133300439300

Offset: 0

Seiichi Manyama, Mar 22 2025

Cf. A129527, A382336.

Cf. A223142, A382333.

G.f. A(x) satisfies A(x) = ( A(x^2)^2 + 4*x/(1-x)^2 )^(1/2).

A382336 Expansion of ( 1 + 9 * Sum_{k>=0} x^(2^k)/(1 - x^(2^k))^2 )^(1/3).

Original entry on oeis.org

1, 3, 0, 0, 21, -111, 504, -2004, 7092, -21150, 43614, 24288, -949878, 7022118, -38308320, 175670820, -691787607, 2250673143, -4994247456, -2841846468, 120496073523, -931900270923, 5282041372722, -25033533979260, 101401747872534, -337523450786736, 757180705527738

Offset: 0

Seiichi Manyama, Mar 22 2025

Cf. A129527, A382335.

Cf. A223143, A382334.

G.f. A(x) satisfies A(x) = ( A(x^2)^3 + 9*x/(1-x)^2 )^(1/3).

A328154 G.f. A(x) satisfies: A(x) = A(x^2) + x / (1 + x)^2.

Original entry on oeis.org

1, -1, 3, -5, 5, -3, 7, -13, 9, -5, 11, -15, 13, -7, 15, -29, 17, -9, 19, -25, 21, -11, 23, -39, 25, -13, 27, -35, 29, -15, 31, -61, 33, -17, 35, -45, 37, -19, 39, -65, 41, -21, 43, -55, 45, -23, 47, -87, 49, -25, 51, -65, 53, -27, 55, -91, 57, -29, 59, -75, 61, -31, 63, -125, 65

Offset: 1

Ilya Gutkovskiy, Oct 05 2019

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

Cf. A000010, A001511, A062570, A088705, A129527.

a:=[1]; for k in [1..65] do if IsOdd(k) then a[k]:=k; else a[k]:=a[k div 2]-k; end if;  end for; a; // Marius A. Burtea, Oct 07 2019

nmax = 65; CoefficientList[Series[Sum[x^(2^k)/(1 + x^(2^k))^2, {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x] // Rest
nmax = 65; CoefficientList[Series[Sum[(-1)^(k + 1) EulerPhi[2 k] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := If[EvenQ[n], a[n/2] - n, n]; Table[a[n], {n, 1, 65}]

a(n) = if (n==1, 1, if (n % 2, n, a(n/2) - n)); \\ Michel Marcus, Oct 07 2019

a(n) = 3*(n>>valuation(n,2)) - n<<1; \\ Kevin Ryde, Oct 06 2020

G.f.: Sum_{k>=0} x^(2^k) / (1 + x^(2^k))^2.
G.f.: Sum_{k>=1} (-1)^(k + 1) * phi(2*k) * x^k / (1 - x^k), where phi = A000010.
a(2*n) = a(n) - 2*n, a(2*n+1) = 2*n + 1.
From Werner Schulte, Oct 05 2020: (Start)
Multiplicative with a(2^e) = 3 - 2^(e+1) and a(p^e) = p^e for e >= 0 and prime p > 2.
Dirichlet g. f.: Sum_{n>0} a(n)/n^s = zeta(s-1) * (1-3/(2^s-1)). (End)

A328969 Irregular table T(n,k), n >= 2, k=1..pi(n). arising in expressing the sequence A006022 as the coefficients depending on the maximal k-th prime factor pk of the formula for A006022(n) of its unique prime factor equation.

Original entry on oeis.org

1, 0, 1, 3, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 1, 7, 0, 0, 0, 0, 4, 0, 0, 5, 0, 1, 0, 0, 0, 0, 0, 1, 9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 7, 0, 0, 1, 0, 0, 0, 5, 1, 0, 0, 0, 15, 0, 0, 0, 0, 0

Offset: 2

Jonathan Blanchette, Nov 01 2019

The length of the n-th row is pi(n) (A000720), i.e., 1,2,2,3,... for n>2.
The sum of the rows equals the sequence A006022.
When n is prime the entire row is 0 except at p=n where T(p,p)=1.

			First few rows are:
  1;
  0, 1;
  3, 0;
  0, 0, 1;
  3, 1, 0;
  0, 0, 0, 1;
  7, 0, 0, 0;
  0, 4, 0, 0;
  5, 0, 1, 0;
  0, 0, 0, 0, 1;
  ...
Examples (see the p_k formulas)
T(2^3,1) = (2^3-1) / (2-1) = 7
T(3^2,1) = (3^2-1) / (3-1) = 4
T(3*2,2) = (6/(2*3)) * (3^2-1) / (3-1) = 4
T(12,1) = (12/(2^2)) * (2^2-1) / (2-1) = 9
T(12,2) = (12/(2^2*3)) * (3-1) / (3-1) = 1
T(15,2) = (15/3) * (3-1) / (3-1) = 5
T(15,3) = (15/(2^2*3)) * (3-1) / (3-1) = 1
T(2*3*5^2*7,3) = (2*3*5^2*7/(2*3*5^2)) * (5^2-1) / (5-1) = 42

Jonathan Blanchette and Robert Laganière, A Curious Link Between Prime Numbers, the Maundy Cake Problem and Parallel Sorting, arXiv:1910.11749 [cs.DS], 2019.

The rows sum to A006022. Cf. A129527 (first column).

Let p_k be the k-th prime, where k is the column index, p_k <= n, and n >= 2, and m_k is the multiplicity of p_k occurring in n:
T(n,p_k) = n * 1/(p_1^m_1*p_2^m_2*...*p_k^m_k) * (p_k^m_k-1)/(p_k-1), if p_k divides n;
T(n,p_k) = 0; if p_k does not divide n.
T(2*n,2) = A129527(n); T(2*n+1,2) = 0.

A335115 a(2n) = 2n - a(n), a(2n+1) = 2n + 1.

Original entry on oeis.org

1, 1, 3, 3, 5, 3, 7, 5, 9, 5, 11, 9, 13, 7, 15, 11, 17, 9, 19, 15, 21, 11, 23, 15, 25, 13, 27, 21, 29, 15, 31, 21, 33, 17, 35, 27, 37, 19, 39, 25, 41, 21, 43, 33, 45, 23, 47, 33, 49, 25, 51, 39, 53, 27, 55, 35, 57, 29, 59, 45, 61, 31, 63, 43, 65, 33, 67, 51, 69, 35, 71, 45, 73, 37, 75

Offset: 1

Ilya Gutkovskiy, May 23 2020

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

Cf. A035263, A050292, A129527, A154269, A001045.

a[n_] := a[n] = If[EvenQ[n], n - a[n/2], n]; Table[a[n], {n, 1, 75}]
nmax = 75; CoefficientList[Series[Sum[(-1)^k x^(2^k)/(1 - x^(2^k))^2, {k, 0, Floor[Log[2, nmax]]}], {x, 0, nmax}], x] // Rest
f[p_, e_] := If[p == 2, (2^(e + 1) + (-1)^e)/3, p^e]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)

a(n) = my(k=valuation(n,2)); (n<<1 + (n>>k)*(-1)^k)/3; \\ Kevin Ryde, Oct 06 2020

G.f.: Sum_{k>=0} (-1)^k * x^(2^k) / (1 - x^(2^k))^2.
G.f. A(x) satisfies: A(x) = x / (1 - x)^2 - A(x^2).
Dirichlet g.f.: zeta(s-1) / (1 + 2^(-s)).
a(n) = Sum_{d|n} A154269(n/d) * d.
Sum_{k=1..n} a(k) ~ 2*n^2/5. - Vaclav Kotesovec, Jun 11 2020
Multiplicative with a(2^e) = A001045(e+1) and a(p^e) = p^e for e >= 0 and prime p > 2. - Werner Schulte, Oct 05 2020

A338046 G.f.: Sum_{k>=0} x^(2^k) / (1 - x^(2^k))^4.

Original entry on oeis.org

1, 5, 10, 25, 35, 66, 84, 145, 165, 255, 286, 430, 455, 644, 680, 961, 969, 1305, 1330, 1795, 1771, 2310, 2300, 3030, 2925, 3731, 3654, 4704, 4495, 5640, 5456, 6945, 6545, 8109, 7770, 9741, 9139, 11210, 10660, 13275, 12341, 15015, 14190, 17490, 16215, 19596, 18424, 22630

Offset: 1

Ilya Gutkovskiy, Oct 08 2020

nonn

Cf. A000292, A000335, A000447, A001511, A002492, A129527, A209229, A328407, A338045.

nmax = 48; CoefficientList[Series[Sum[x^(2^k) /(1 - x^(2^k))^4, {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x] // Rest
a[n_] := If[EvenQ[n], a[n/2] + n (n + 1) (n + 2)/6, n (n + 1) (n + 2)/6]; Table[a[n], {n, 1, 48}]
Table[(1/6) DivisorSum[n, Boole[IntegerQ[Log[2, n/#]]] # (# + 1) (# + 2) &], {n, 1, 48}]

G.f. A(x) satisfies: A(x) = A(x^2) + x / (1 - x)^4.
a(2*n) = a(n) + A002492(n), a(2*n+1) = A000447(n+1).
a(n) = (1/6) * Sum_{d|n} A209229(n/d) * d * (d + 1) * (d + 2).
Product_{n>=1} (1 + x^n)^a(n) = g.f. for A000335.

A373186 Expansion of Sum_{k>=0} x^(3^k) / (1 - x^(3^k))^3.

Original entry on oeis.org

1, 3, 7, 10, 15, 24, 28, 36, 52, 55, 66, 88, 91, 105, 135, 136, 153, 195, 190, 210, 259, 253, 276, 336, 325, 351, 430, 406, 435, 520, 496, 528, 627, 595, 630, 754, 703, 741, 871, 820, 861, 1008, 946, 990, 1170, 1081, 1128, 1312, 1225, 1275, 1479, 1378, 1431, 1680, 1540

Offset: 1

Seiichi Manyama, May 27 2024

nonn

Cf. A129527, A373187.
Cf. A327625, A338045.
Cf. A000217.

G.f. A(x) satisfies A(x) = x/(1 - x)^3 + A(x^3).

a(3*n+1) = A000217(3*n+1), a(3*n+2) = A000217(3*n+2) and a(3*n+3) = A000217(3*n+3) + a(n+1) for n >= 0.

A373397 Expansion of Sum_{k>=0} x^(6^k) / (1 - x^(6^k))^2.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 7, 8, 9, 10, 11, 14, 13, 14, 15, 16, 17, 21, 19, 20, 21, 22, 23, 28, 25, 26, 27, 28, 29, 35, 31, 32, 33, 34, 35, 43, 37, 38, 39, 40, 41, 49, 43, 44, 45, 46, 47, 56, 49, 50, 51, 52, 53, 63, 55, 56, 57, 58, 59, 70, 61, 62, 63, 64, 65, 77, 67, 68, 69, 70, 71, 86, 73, 74, 75, 76, 77, 91

Offset: 1

Seiichi Manyama, Jun 04 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A129527, A327625, A359099, A359100, A373188.

Cf. A373216.

G.f. A(x) satisfies A(x) = x/(1 - x)^2 + A(x^6).

If n == 0 (mod 6), a(n) = n + a(n/6) otherwise a(n) = n.

A188023 Triangle read by rows, T(n,k) = k*A115361(n-1,k-1).

Original entry on oeis.org

1, 1, 2, 0, 0, 3, 1, 2, 0, 4, 0, 0, 0, 0, 5, 0, 0, 3, 0, 0, 6, 0, 0, 0, 0, 0, 0, 7, 1, 2, 0, 4, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 5, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 3, 0, 0, 6, 0, 0, 0, 0, 0, 12

Offset: 1

Gary W. Adamson, Mar 19 2011

Triangle obtained by multiplying the lower triangular matrices A115361 and A127648.

			First few rows of the triangle =
1
1, 2
0, 0, 3
1, 2, 0, 4
0, 0, 0, 0, 5
0, 0, 3, 0, 0, 6
0, 0, 0, 0, 0, 0, 7
1, 2, 0, 4, 0, 0, 0, 8
0, 0, 0, 0, 0, 0, 0, 0, 9
0, 0, 0, 0, 5, 0, 0, 0, 0, 10
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11
0, 0, 3, 0, 0, 6, 0, 0, 0, 0, 0, 12
...

Cf. A115361, A127648, A129527 (row sums)

A334070 Number of even-order elements in the multiplicative group of integers modulo n.

Original entry on oeis.org

0, 0, 1, 1, 3, 1, 3, 3, 3, 3, 5, 3, 9, 3, 7, 7, 15, 3, 9, 7, 9, 5, 11, 7, 15, 9, 9, 9, 21, 7, 15, 15, 15, 15, 21, 9, 27, 9, 21, 15, 35, 9, 21, 15, 21, 11, 23, 15, 21, 15, 31, 21, 39, 9, 35, 21, 27, 21, 29, 15, 45, 15, 27, 31, 45, 15, 33, 31, 33, 21, 35, 21, 63

Offset: 1

Robert A. Jones, Apr 13 2020

nonn

The number of even-order elements in a finite abelian group G is |G| - b(|G|), where b is given by A000265. To see this, decompose G as a product of cyclic groups of orders {m_k}. G has [prod_k b(m_k)] elements of odd order, since an element has odd order if and only if all its components have odd order, and each C_m factor has b(m) elements of odd order. Since b can be pulled outside the product, G has b(|G|) elements of odd order. Using that the order of (Z/nZ)^x is phi(n), we obtain a(n) = phi(n) - b(phi(n)).

Since phi(n) is even when n > 2, a(n) is odd when n > 2.

			For n = 10, the elements of (Z_n)^x with even order are 3 (order 4), 7 (order 4), and 9 (order 2). Thus, a(10) = 3.

Cf. A000010, A053575, A129527, A331739 (number of even-order elements in Z_n).

a:= n-> (t-> t-t/2^padic[ordp](t, 2))(numtheory[phi](n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Apr 14 2020

a[n_] := Length@
  Select[Range[n] - 1, EvenQ[MultiplicativeOrder[#, n]] &];
oddPart[n_] := n/2^IntegerExponent[n,2];
a[n_] := EulerPhi[n] - oddPart[EulerPhi[n]];

a(n) = A000010(n) - A053575(n) = A331739(A000010(n)).

cp's OEIS Frontend

A382335 Expansion of ( 1 + 4 * Sum_{k>=0} x^(2^k)/(1 - x^(2^k))^2 )^(1/2).

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A382336 Expansion of ( 1 + 9 * Sum_{k>=0} x^(2^k)/(1 - x^(2^k))^2 )^(1/3).

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A328154 G.f. A(x) satisfies: A(x) = A(x^2) + x / (1 + x)^2.

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A328969 Irregular table T(n,k), n >= 2, k=1..pi(n). arising in expressing the sequence A006022 as the coefficients depending on the maximal k-th prime factor pk of the formula for A006022(n) of its unique prime factor equation.

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A335115 a(2*n) = 2*n - a(n), a(2*n+1) = 2*n + 1.

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A338046 G.f.: Sum_{k>=0} x^(2^k) / (1 - x^(2^k))^4.

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Mathematica

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A373186 Expansion of Sum_{k>=0} x^(3^k) / (1 - x^(3^k))^3.

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A373397 Expansion of Sum_{k>=0} x^(6^k) / (1 - x^(6^k))^2.

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A188023 Triangle read by rows, T(n,k) = k*A115361(n-1,k-1).

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A334070 Number of even-order elements in the multiplicative group of integers modulo n.

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A335115 a(2n) = 2n - a(n), a(2n+1) = 2n + 1.