A048272 -id:A048272 - cp's OEIS frontend

A307680 Expansion of e.g.f. Product_{k>=1} (1 + x^k/(1 - x)^k)^(1/k).

1, 1, 3, 17, 131, 1239, 14029, 187627, 2906553, 50982929, 993806531, 21270277401, 496425262123, 12577053063847, 344382608381421, 10139294386051139, 319175215666010609, 10684742192933940897, 378662321114852778883, 14158327369578651838369, 557151639159864934384851

Offset: 0

Ilya Gutkovskiy, Apr 21 2019

nonn

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 131*x^4/4! + 1239*x^5/5! + 14029*x^6/6! + 187627*x^7/7! + 2906553*x^8/8! + ...

Cf. A048272, A129519, A168243, A307679, A320564.

nmax = 20; CoefficientList[Series[Product[(1 + x^k/(1 - x)^k)^(1/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[Sum[Sum[(-1)^(d + 1), {d, Divisors[k]}] x^k/(k (1 - x)^k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: exp(Sum_{k>=1} A048272(k)*x^k/(k*(1 - x)^k)).

a(n) = Sum_{k=0..n} binomial(n-1,k-1)*A168243(k)*n!/k!.

A325941 Expansion of Sum_{k>=1} k * x^(2*k) / (1 + x^k)^2.

Original entry on oeis.org

0, 1, -2, 5, -4, 4, -6, 17, -14, 6, -10, 28, -12, 8, -36, 49, -16, 13, -18, 46, -52, 12, -22, 100, -44, 14, -68, 64, -28, 24, -30, 129, -84, 18, -92, 121, -36, 20, -100, 166, -40, 32, -42, 100, -192, 24, -46, 292, -90, 31, -132, 118, -52, 40, -148, 232, -148, 30, -58, 264

Offset: 1

Ilya Gutkovskiy, Sep 09 2019

sign

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

Cf. A000593, A048272, A094471, A143520.

nmax = 60; CoefficientList[Series[Sum[k x^(2 k)/(1 + x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[(-1)^(n/d) (n - d), {d, Divisors[n]}], {n, 1, 60}]

{a(n) = sumdiv(n, d, (-1)^(n/d)*(n-d))} \\ Seiichi Manyama, Sep 14 2019

G.f.: Sum_{k>=2} (-1)^k * (k - 1) * x^k / (1 - x^k)^2.
a(n) = Sum_{d|n} (-1)^(n/d) * (n - d).
a(n) = A000593(n) - n * A048272(n).

A329393 Number of odd divisors minus number of even divisors of the n-th composite.

Original entry on oeis.org

-1, 0, -2, 3, 0, -2, 0, 4, -3, 0, -2, 4, 0, -4, 3, 0, 4, -2, 0, -4, 4, 0, 4, -3, 0, 4, -4, 0, -2, 6, 0, -6, 3, 0, 4, -2, 0, 4, -4, 4, 0, -4, 0, 6, -5, 4, 0, -2, 4, 0, -6, 0, 6, -2, 4, 0, -6, 5, 0, -4, 4, 0, 4, -4, 0, 4, -2, 4, 0, 4, -8, 0, 6

Offset: 1

Enrique Navarrete, Nov 12 2019

sign

The mode, or most frequent value of this sequence is 0, which corresponds to composites with equal number of odd and even divisors, A016825(n), n >= 1. The next most frequent value is 4.

The value 2 does not appear in this sequence, in contrast to A048272, where A048272(p)=2 for every p = odd prime.

			a(1) = -1 since the first composite number is 4, which has 1 odd divisor (1), and 2 even divisors (2,4).
a(2) = 0 since the second composite number is 6, which has 2 odd divisors (1,3) and 2 even divisors (2,6).

Harvey P. Dale, Table of n, a(n) for n = 1..3000

Cf. A002808, A016825, A048272.

f[p_, e_] := If[p == 2, 1 - e, 1 + e]; diffNum[1] = 1; diffNum[n_] := Times @@ (f @@@ FactorInteger[n]); diffNum /@ Select[Range[100], CompositeQ] (* Amiram Eldar, Nov 25 2019 *)
odmed[n_]:=With[{divs=Divisors[n]},Count[divs,?OddQ]-Count[divs,?EvenQ]]; odmed/@Select[Range[100],CompositeQ] (* Harvey P. Dale, Nov 18 2024 *)

A368744 a(n) = Sum_{d|n} (-1)^(d+1)*phi(d), where phi(n) = A000010(n).

Original entry on oeis.org

1, 0, 3, -2, 5, 0, 7, -6, 9, 0, 11, -6, 13, 0, 15, -14, 17, 0, 19, -10, 21, 0, 23, -18, 25, 0, 27, -14, 29, 0, 31, -30, 33, 0, 35, -18, 37, 0, 39, -30, 41, 0, 43, -22, 45, 0, 47, -42, 49, 0, 51, -26, 53, 0, 55, -42, 57, 0, 59, -30, 61, 0, 63, -62, 65, 0, 67, -34, 69, 0, 71, -54, 73, 0, 75

Offset: 1

Peter Bala, Jan 21 2024

Recall Gauss's identity Sum_{d|n} phi(d) = n.

a(n) is a multiplicative function of n since both (-1)^(n+1) and phi(n) are multiplicative functions of n.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A000010, A006519, A048272, A321543.

with(numtheory): seq( add( (-1)^(d+1)*phi(d), d in divisors(n)), n = 1..75);

A368744[n_] := DivisorSum[n, (-1)^(#+1)*EulerPhi[#]&];
Array[A368744, 100] (* Paolo Xausa, Jan 30 2024 *)
a[n_] := (2^(1-IntegerExponent[n, 2]) - 1) * n ; Array[a, 100] (* Amiram Eldar, Jan 31 2024 *)

a(n) = sumdiv(n, d, (-1)^(d+1)*eulerphi(d)); \\ Michel Marcus, Jan 30 2024

PARI
```
a(n) = (2/(1<Amiram Eldar, Jan 31 2024
```

def A368744(n): return ((n<<1)>>(~n & n-1).bit_length())-n # Chai Wah Wu, Jan 30 2024

a(n) = -Sum_{k = 1..n} (-1)^(lcm(k, n)/k) = -Sum_{k = 1..n} (-1)^(n/gcd(k, n)).
a(2*n+1) = 2*n + 1; a(4*n+2) = 0.
Multiplicative: a(2^k) = 2 - 2^k and for odd prime p, a(p^k) = p^k.
Dirichlet g.f.: (1 - 3/2^s)/(1 - 1/2^s) * zeta(s-1).
From Amiram Eldar, Jan 31 2024: (Start)
a(n) = (2/A006519(n) - 1) * n.
Sum_{k=1..n} a(k) ~ n^2/6. (End)

A369100 Dirichlet g.f.: zeta(s)^3 * (1 - 2^(1-s))^2.

Original entry on oeis.org

1, -1, 3, -2, 3, -3, 3, -2, 6, -3, 3, -6, 3, -3, 9, -1, 3, -6, 3, -6, 9, -3, 3, -6, 6, -3, 10, -6, 3, -9, 3, 1, 9, -3, 9, -12, 3, -3, 9, -6, 3, -9, 3, -6, 18, -3, 3, -3, 6, -6, 9, -6, 3, -10, 9, -6, 9, -3, 3, -18, 3, -3, 18, 4, 9, -9, 3, -6, 9, -9, 3, -12, 3, -3, 18

Offset: 1

Vaclav Kotesovec, Jan 13 2024

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

Cf. A007425, A048272, A288571.

Table[Sum[Sum[-(-1)^d, {d, Divisors[k]}]*(-1)^(n/k+1), {k, Divisors[n]}], {n, 1, 100}]
f[p_, e_] := (e + 1)*(e + 2)/2; f[2, e_] := (e^2 - 5*e + 2)/2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 13 2024 *)

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e=f[i,2]; if(p == 2, (e^2-5*e+2)/2, (e+1)*(e+2)/2));} \\ Amiram Eldar, Jan 13 2024

Sum_{k=1..n} a(k) ~ n * log(2)^2.

Multiplicative with a(2^e) = (e^2-5*e+2)/2, and a(p^e) = (e+1)*(e+2)/2 for an odd prime p. - Amiram Eldar, Jan 13 2024

A373275 a(n) = Sum_{d|n} (-1)^(d-1) * 2^(n/d-1).

Original entry on oeis.org

1, 1, 5, 5, 17, 29, 65, 117, 261, 497, 1025, 2017, 4097, 8129, 16405, 32629, 65537, 130845, 262145, 523765, 1048645, 2096129, 4194305, 8386641, 16777233, 33550337, 67109125, 134209477, 268435457, 536855053, 1073741825, 2147450741, 4294968325, 8589869057

Offset: 1

Seiichi Manyama, May 29 2024

Cf. A048272, A373276.

Cf. A034729, A321386.

a(n) = sumdiv(n, d, (-1)^(d-1)*2^(n/d-1));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (2*x)^k/(1+x^k))/2)

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

If p is an odd prime, a(p) = 1 + 2^(p-1).

A373276 a(n) = Sum_{d|n} (-1)^(d-1) * 3^(n/d-1).

Original entry on oeis.org

1, 2, 10, 23, 82, 236, 730, 2156, 6571, 19604, 59050, 176918, 531442, 1593596, 4783060, 14346689, 43046722, 129133838, 387420490, 1162241726, 3486785140, 10460294156, 31381059610, 94143003584, 282429536563, 847288078004, 2541865834900, 7625595889958

Offset: 1

Seiichi Manyama, May 29 2024

Cf. A048272, A373275.

Cf. A321386, A357051.

a(n) = sumdiv(n, d, (-1)^(d-1)*3^(n/d-1));

my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, (3*x)^k/(1+x^k))/3)

G.f.: 1/3 * Sum_{k>=1} (3*x)^k / (1 + x^k).

If p is an odd prime, a(p) = 1 + 3^(p-1).

A304907 Expansion of x * (d/dx) 1/(1 - Sum_{k>=1} x^k/(1 + x^k)).

Original entry on oeis.org

0, 1, 2, 9, 16, 35, 84, 161, 312, 639, 1240, 2354, 4536, 8593, 16128, 30360, 56672, 105213, 195174, 360582, 664040, 1220730, 2238324, 4095035, 7479552, 13636750, 24821108, 45114813, 81887008, 148438211, 268763160, 486082263, 878200416, 1585098372, 2858378368, 5149986275

Offset: 0

Ilya Gutkovskiy, May 20 2018

nonn

Sum of all parts of all Carlitz compositions (compositions without adjacent equal parts) of n.

Index entries for sequences related to compositions

Cf. A001787, A003242, A048272, A066186, A066189.

nmax = 35; CoefficientList[Series[x D[1/(1 - Sum[x^k/(1 + x^k), {k, 1, nmax}]), x], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[-(-1)^d, {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[n a[n], {n, 0, 35}]

a(n) = n*A003242(n).

A369101 Dirichlet g.f.: zeta(s-3)^2 * (1 - 2^(4-s)) / zeta(s).

Original entry on oeis.org

1, -1, 53, -64, 249, -53, 685, -960, 2133, -249, 2661, -3392, 4393, -685, 13197, -11264, 9825, -2133, 13717, -15936, 36305, -2661, 24333, -50880, 46625, -4393, 76545, -43840, 48777, -13197, 59581, -118784, 141033, -9825, 170565, -136512, 101305, -13717, 232829

Offset: 1

Vaclav Kotesovec, Jan 13 2024

In general, for k > 0, if Dirichlet g.f. is zeta(s-k)^2 * (1 - 2^(k+1-s)) / zeta(s), then a(n) ~ log(2) * n^(k+1) / ((k+1) * zeta(k+1)).

Cf. A048272 (k=0), A332794 (k=1), A368929 (k=2).

Cf. A000578, A059376.

Table[Sum[DivisorSum[k, #^3*MoebiusMu[k/#]&]*(-1)^(n/k+1)*(n/k)^3, {k, Divisors[n]}], {n, 1, 50}]
f[p_, e_] := p^(3*e-3) * (1 + (e+1)*(p^3-1)); f[2, e_] := -(7*e-6)*8^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 13 2024 *)

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e=f[i,2]; if(p == 2, -(7*e-6)*8^(e-1), p^(3*e-3) * (1 + (e+1)*(p^3-1))));} \\ Amiram Eldar, Jan 13 2024

Sum_{k=1..n} a(k) ~ 45 * log(2) * n^4 / (2*Pi^4).

Multiplicative with a(2^e) = -(7*e-6)*8^(e-1), and a(p^e) = p^(3*e-3) * (1 + (e+1)*(p^3-1)) for an odd prime p. - Amiram Eldar, Jan 13 2024

cp's OEIS Frontend

A307680 Expansion of e.g.f. Product_{k>=1} (1 + x^k/(1 - x)^k)^(1/k).

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A325941 Expansion of Sum_{k>=1} k * x^(2*k) / (1 + x^k)^2.

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A329393 Number of odd divisors minus number of even divisors of the n-th composite.

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A368744 a(n) = Sum_{d|n} (-1)^(d+1)*phi(d), where phi(n) = A000010(n).

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A369100 Dirichlet g.f.: zeta(s)^3 * (1 - 2^(1-s))^2.

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A373275 a(n) = Sum_{d|n} (-1)^(d-1) * 2^(n/d-1).

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A373276 a(n) = Sum_{d|n} (-1)^(d-1) * 3^(n/d-1).

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PARI

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A304907 Expansion of x * (d/dx) 1/(1 - Sum_{k>=1} x^k/(1 + x^k)).

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