A222171 -id:A222171 - cp's OEIS frontend

A113262 One quarter of the number of solutions to a^2 + b^2 + 3c^2 + 3d^2 = n.

1, 1, 1, 5, 6, 1, 8, 13, 1, 6, 12, 5, 14, 8, 6, 29, 18, 1, 20, 30, 8, 12, 24, 13, 31, 14, 1, 40, 30, 6, 32, 61, 12, 18, 48, 5, 38, 20, 14, 78, 42, 8, 44, 60, 6, 24, 48, 29, 57, 31, 18, 70, 54, 1, 72, 104, 20, 30, 60, 30, 62, 32, 8, 125, 84, 12, 68, 90, 24, 48, 72, 13, 74, 38, 31

Offset: 1

Michael Somos, Oct 21 2005

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 223, Entry 3(iv).
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 229.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 79, Eq. (32.3), p. 76, Eq. (31.43).

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A034896, A222171.

A034896[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, q]*EllipticTheta[3, 0, q^3])^2, {q, 0, n}]; Table[A034896[n]/4, {n, 1, 50}] (* G. C. Greubel, Dec 24 2017 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, # KroneckerSymbol[ 9, #] (-1)^(n + #) &]]; (* Michael Somos, Nov 10 2018 *)

{a(n) = if( n<1, 0, sumdiv(n, d, d * kronecker(9, d) * (-1)^(n-d)))};

{a(n) = my(A, p, e); if(n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==3, 1, (p^(e+1) - 1) / (p - 1) - 2*(p==2))))};

a(n) is multiplicative with a(3^e) = 1, a(2^e) = 2^(e+1) - 3, a(p^e) = (p^(e+1) - 1) / (p - 1) if p > 3.
G.f.: Sum_{k>0} k * x^k / (1 - (-x)^k) * Kronecker(9, k) = ((theta_3(x) * theta_3(x^3))^2 - 1) / 4.
A034896(n) = 4*a(n) if n > 0.
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/24 = 0.411233... (A222171). - Amiram Eldar, Dec 01 2022
Dirichlet g.f.: zeta(s)*zeta(s-1)*(4^(1-s)-2^(1-s)+1)*(1-3^(1-s)). - Amiram Eldar, Jan 06 2023

A185152 Expansion of (q/2) * phi(q)^3 (d/dq) phi(q) in powers of q.

Original entry on oeis.org

1, 6, 12, 12, 30, 72, 56, 24, 117, 180, 132, 144, 182, 336, 360, 48, 306, 702, 380, 360, 672, 792, 552, 288, 775, 1092, 1080, 672, 870, 2160, 992, 96, 1584, 1836, 1680, 1404, 1406, 2280, 2184, 720, 1722, 4032, 1892, 1584, 3510, 3312, 2256, 576, 2793, 4650

Offset: 1

Michael Somos, Jan 23 2012

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Ramanujan's Eisenstein series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

			G.f. = x + 6*x^2 + 12*x^3 + 28*x^4 + 30*x^5 + 72*x^6 + 56*x^7 + 24*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000027, A000118, A000290, A046897, A186690, A222171.

Cf. A000122, A000700, A004009, A006352, A010054, A013973, A121373.

a[ n_] := If[ n == 0, 0, n Sum[ d Sign@Mod[d, 4], {d, Divisors@n}]]; (* Michael Somos, Jun 20 2015 *)
a[ n_] := Sign[n] With[ {f = Series[ EllipticTheta[ 3, 0, q], {q, 0, Abs@n + 1}]}, SeriesCoefficient[ (q/2) f^3 D[f, q], Abs@n]]; (* Michael Somos, Jun 20 2015 *)
a[ n_] := Sign[n] With[ {f = Series[ EllipticTheta[ 3, x, q], {q, 0, Abs@n + 1}]}, SeriesCoefficient[ (-1/8) f^3 D[f, x, x] /. x -> 0, Abs@n]]; (* Michael Somos, Jun 20 2015 *)

{a(n) = if( n==0, 0, n * sumdiv( n, d, if( d%4, d)))};

Expansion of (-1/8) * theta_3(0,q)^3 * theta_3(0,q)'' in powers of nome q.
Expansion of (-1/24) * q * (d/dq) (P(q) - 4 * P(q^4)) where P() is a Ramanujan Eisenstein series.
Expansion of (1/8) * (E(k^2) - (1-k^2) * K(k^2)) * K(k^2)^3 / (Pi/2)^4 in powers of nome q.
Multiplicative with a(2^e) = 3 * 2^e if e>0, a(p^e) = p^e * (p^(e+1) - 1) / (p - 1) otherwise.
G.f.: Sum_{k>0} k^2 * x^k / (1 + (-x)^k)^2.
G.f.: Sum_{k>0} k^2 * x^k / (1 - x^k)^2 * (mod(k, 4) > 0).
a(n) = n * Sum of divisors of n that are not divisible by 4 = n * A046897(n).
a(n) = - a(-n). for all n in Z. Convolution of A000118 and A186690.
From Amiram Eldar, Sep 12 2023: (Start)
Dirichlet g.f.: (1 - 1/2^(2*s-4)) * zeta(s-2) * zeta(s-1).
Sum_{k=1..n} a(k) ~ c * n^3, where c = Pi^2/24 = 0.41123... (A222171) . (End)

A369180 Alternating sum of the k-adic valuations (ruler functions) of n.

Original entry on oeis.org

0, 1, -1, 3, -1, 1, -1, 5, -3, 1, -1, 4, -1, 1, -3, 8, -1, 0, -1, 4, -3, 1, -1, 7, -3, 1, -5, 4, -1, 1, -1, 10, -3, 1, -3, 5, -1, 1, -3, 7, -1, 1, -1, 4, -6, 1, -1, 11, -3, 0, -3, 4, -1, -1, -3, 7, -3, 1, -1, 6, -1, 1, -6, 14, -3, 1, -1, 4, -3, 1, -1, 9, -1, 1, -6, 4, -3, 1

Offset: 1

Friedjof Tellkamp, Jan 15 2024

sign

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A007814, A007949, A222171, A235127, A309891.

a:= n-> add((-1)^i*padic[ordp](n, i), i=2..n):
seq(a(n), n=1..78);  # Alois P. Heinz, Jan 15 2024

z = 70; Sum[(-1)^k IntegerExponent[Range[z], k], {k, 2, z}]

a(n) = sum(k=2, n, (-1)^k * valuation(n,k)); \\ Michel Marcus, Jan 18 2024

a(n)=sumdiv(n,k, if(k>1, (-1)^k * valuation(n, k))) \\ Charles R Greathouse IV, Jan 23 2025

a(n) = Sum_{k=2..n} (-1)^k * valuation(n,k).
a(n) = A007814(n) - A007949(n) + A235127(n) - (...).
G.f.: Sum_{k>=2, j>=1} (-1)^k x^(k^j)/(1-x^(k^j)).
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{n=1..m} a(n) = log(2).
Dirichlet g.f.: zeta(s) * Sum_{k>=1} (1 - eta(ks)).
Sum_{n>=1} a(n)/n^2 = Pi^2/24.

A276712 Decimal expansion of zeta(3)/8.

Original entry on oeis.org

1, 5, 0, 2, 5, 7, 1, 1, 2, 8, 9, 4, 9, 4, 9, 2, 8, 5, 6, 7, 4, 9, 6, 7, 2, 7, 0, 1, 8, 8, 9, 3, 1, 2, 4, 8, 8, 4, 5, 6, 2, 3, 2, 8, 6, 5, 4, 2, 5, 6, 2, 3, 6, 0, 2, 2, 4, 0, 3, 3, 9, 4, 4, 4, 1, 7, 7, 2, 9, 7, 7, 5, 7, 2, 3, 2, 8, 9

Offset: 0

Terry D. Grant, Sep 15 2016

			0.150257112894949285674967270188...

James Dodson, The Mathematical Repository Containing Analytical Solutions of Five Hundred Questions: Mostly Selected from Scarce and Valuable Authors, (1748), page 375.

G. C. Greubel, Table of n, a(n) for n = 0..5000
R. Barbieri, J. A. Mignaco and E. Remiddi, Electron form factors up to fourth order. I., Il Nuovo Cim. 11A (4) (1972) 824-864, table II (5)
Nick Lord, Problen 89.D, Problem Corner, The Mathematical Gazette, Vol. 89, No. 514 (2005), pp. 115-119; Solution, ibid., Vol. 89, No. 516 (2005), pp. 539-542.
Michael Penn, The solution is an important constant, YouTube video, 2021.

Cf. A001008, A002117, A002805, A013661, A111003, A222171, A233091.

SetDefaultRealField(RealField(120)); L:=RiemannZeta();  Evaluate(L,3)/8; // G. C. Greubel, Nov 24 2021

Mathematica
```
RealDigits[(Zeta[3])/8, 10, 100][[1]]
```

zeta(3)/8 \\ Michel Marcus, Sep 16 2016

Sage
```
(zeta(3)/8).n(100)
```

Equals Sum_{n>=1} 1/(2n)^3 = 1/8 + 1/64 + 1/216 + 1/512 + ...
Equals A002117/8.
zeta(3)/8 + A233091 = Sum_{n>=1} 1/(2n+1)^3 + Sum_{n>=1} 1/(2n)^3 = zeta(3).
Equals Sum_{k>=1} (-1)^(k+1) * H(k)/(k+1)^2, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Jul 22 2020
Equals Integral_{x=0..Pi/4} log(sin(x))*log(cos(x))/(sin(x)*cos(x)) dx (Lord, 2005). - Amiram Eldar, Jun 23 2023
Equals -integral_{x=0..1} log(x) log(1+x)/(1+x). [Barbieri] - R. J. Mathar, Jun 07 2024

A228274 a(n) = Sum_{d|n, n/d odd} n * d.

Original entry on oeis.org

1, 4, 12, 16, 30, 48, 56, 64, 117, 120, 132, 192, 182, 224, 360, 256, 306, 468, 380, 480, 672, 528, 552, 768, 775, 728, 1080, 896, 870, 1440, 992, 1024, 1584, 1224, 1680, 1872, 1406, 1520, 2184, 1920, 1722, 2688, 1892, 2112, 3510, 2208, 2256, 3072, 2793, 3100

Offset: 1

Michael Somos, Aug 19 2013

			G.f. = x + 4*x^2 + 12*x^3 + 16*x^4 + 30*x^5 + 48*x^6 + 56*x^7 + 64*x^8 + ...
a(6) = 48 = 6 * (2 + 6). a(9) = 117 = 9 * (1 + 3 + 9). a(10) = 120 = 10 * (2 + 10).

G. C. Greubel, Table of n, a(n) for n = 1..10000
M. A. Basoco, On the Fourier developments of a certain class of theta quotients, Bull. Amer. Math. Soc. 49 (1943), 299-306.

Cf. A002131, A007331, A222171.

A228274[n_] := If[ n < 1, 0, n Sum[ d Mod[n / d, 2], {d, Divisors @ n}]]; Table[A228274[n], {n, 50}]

{a(n) = if( n<1, 0, n * sumdiv(n, d, d * (n/d % 2)))};

Multiplicative with a(2^e) = 4^e, a(p^e) = p^e * (p^(e+1) - 1) / (p - 1) if p>2.
G.f.: Sum_{k>0} k^2 * (x^k + x^(3*k)) / (1 - x^(2*k))^2. [see Basoco (1943) bottom page 305]
G.f.: Sum_{k>0} k^2 * (3 - (-1)^k)/4 * x^k / (1 - x^k)^2.
G.f.: Sum_{k>0 odd} k * (x^k + x^(2*k)) / (1 - x^k)^3.
a(n) = n * A002131(n). a(2*n) = 4 * a(n).
a(n) = A007331(n) - 4 * Sum_{k>0} A002131(k) * A002131(n-k). [see Basoco (1943) page 305 equation (9)]
Sum_{k=1..n} a(k) ~ c * n^3, where c = Pi^2/24 = 0.411233... (A222171). - Amiram Eldar, Nov 30 2022

A282468 Decimal expansion of the zeta function at 2 of every second prime number.

Original entry on oeis.org

1, 4, 4, 7, 1, 5, 5, 8, 6, 6, 8, 8, 7

Offset: 0

Terry D. Grant, Apr 14 2017

From Husnain Raza, Aug 30 2023: (Start)
Note that since p_n > n*log(n), we can place a bound on the tail of the sum:
Sum_{n >= N} (prime(2n))^(-2) <= Sum_{n >= N} (2*n*log(2n))^(-2) <= Integral_{x=N..oo} (2*x*log(2x))^(-2) dx.
Taking the sum over all primes < 10^12, we see that the constant lies between 0.14471558668870 and 0.14471558668873. (End)

			1/3^2 + 1/7^2 + 1/13^2 + 1/19^2 + 1/29^2 + ... = 0.14471558...

Husnain Raza, PARI program for calculating more digits.

Cf. A031215, A085548.

Zeta functions at 2: A085548 (for primes), A275647 (for nonprimes), A013661 (for natural numbers), A117543 (for semiprimes), A131653 (for triprimes), A222171 (for even numbers), A111003 (for odd numbers).

PARI
```
sum(n=1, 2500000, 1./prime(2*n)^2)
```
PARI
```
\\ see Raza link
```

Equals Sum_{n>=1} 1/A031215(n)^2 = Sum_{n>=1} 1/prime(2n)^2.

a(8)-a(12) from Husnain Raza, Aug 31 2023

A343481 a(n) is the sum of all digits of n in every prime base 2 <= p <= n.

Original entry on oeis.org

1, 3, 3, 6, 6, 10, 11, 11, 10, 15, 16, 22, 21, 21, 23, 30, 32, 40, 42, 42, 39, 48, 52, 53, 49, 52, 53, 63, 66, 77, 83, 82, 76, 77, 82, 94, 87, 85, 90, 103, 107, 121, 123, 129, 120, 135, 144, 147, 153, 150, 151, 167, 176, 178, 185, 181, 168, 185, 194, 212, 199

Offset: 2

Amiram Eldar, Apr 16 2021

			a(5) = 6 since in the prime bases 2, 3 and 5 the representations of 5 are 101_2, 12_3 and 10_5, respectively, and (1 + 0 + 1) + (1 + 2) + (1 + 0) = 6.

Amiram Eldar, Table of n, a(n) for n = 2..10001
Robin Fissum, Digit sums and the number of prime factors of the factorial n!=1.2...n, Bachelor's project in BMAT, Norwegian University of Science and Technology, 2020; ResearchGate link.

Cf. A014837, A043306, A072691, A073002, A222171.

s[n_, b_] := Plus @@ IntegerDigits[n, b]; ps[n_] := Select[Range[n], PrimeQ]; a[n_] := Sum[s[n, b], {b, ps[n]}]; Array[a, 100, 2]

a(n) = sum(b=2, n, if (isprime(b), sumdigits(n, b))); \\ Michel Marcus, Apr 17 2021

a(n) ~ (1-Pi^2/12)*n^2/log(n) + c*n^2/log(n)^2 + o(n^2/log(n)^2), where c = 1 - Pi^2/24 + zeta'(2)/2 = 1 - A222171 - (1/2)*A073002 = 0.1199923561... (Fissum, 2020).

A358345 a(n) is the number of even square divisors of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Nov 11 2022

nonn

First differs from A235127 at n = 36.

The first position of k >= 0 in this sequence is A187941(k)^2.

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

Cf. A016742, A046951, A187941, A222171, A235127, A298735.

f1[p_, e_] := Floor[e/2] + 1; f2[p_, e_] := If[p == 2, 1, Floor[e/2] + 1]; a[1] = 0; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i=1, #f~, 1+f[i,2]\2) - prod(i=1, #f~, if(f[i,1] == 2, 1, 1+f[i,2]\2))};

a(n) = sumdiv(n, d, if (issquare(d) && !(d%2), 1)); \\ Michel Marcus, Nov 11 2022

a(n) = A046951(n) - A298735(n).
a(n) = 2 * A046951(n) - A046951(4*n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi^2/24 (A222171).

A348372 Decimal expansion of Sum_{k>=2} H(k)*H(k+1)/(k^3-k), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

Original entry on oeis.org

8, 8, 6, 7, 0, 9, 5, 8, 0, 1, 2, 8, 3, 4, 9, 1, 0, 5, 4, 8, 2, 1, 5, 8, 0, 4, 6, 8, 2, 7, 0, 4, 3, 7, 1, 1, 9, 3, 0, 2, 7, 6, 2, 3, 2, 3, 5, 7, 8, 0, 1, 5, 0, 8, 7, 7, 3, 8, 3, 8, 8, 8, 7, 3, 1, 5, 6, 5, 9, 9, 2, 6, 6, 1, 2, 8, 8, 6, 6, 9, 1, 3, 5, 5, 1, 3, 6, 9, 0, 1, 2, 3, 5, 7, 2, 5, 0, 6, 5, 5, 2, 7, 7, 9, 9

Offset: 0

Amiram Eldar, Oct 15 2021

			0.88670958012834910548215804682704371193027623235780...

Ovidiu Furdui and Alina Sȋntămărian, Problem 12060, The American Mathematical Monthly, Vol. 125, No. 7 (2018), p. 661; Summation by Parts and an Euler Sum, Solution to Problem 12060 by Omran Kouba, ibid., Vol. 127, No. 3 (2020), pp. 274-282.

Cf. A001008, A002805, A002117, A222171.

RealDigits[5/2 - Pi^2/24 - Zeta[3], 10, 100][[1]]

Equals 5/2 - Pi^2/24 - zeta(3).

A348731 Decimal expansion of Integral_{x=0..1} x*log(x)/(1+x+x^2) dx (negated).

Original entry on oeis.org

1, 5, 7, 6, 6, 0, 1, 4, 9, 1, 6, 7, 8, 3, 2, 3, 3, 0, 3, 9, 0, 5, 4, 4, 6, 7, 4, 0, 6, 9, 9, 6, 2, 2, 1, 8, 2, 2, 3, 7, 4, 9, 4, 6, 5, 4, 6, 2, 9, 5, 6, 7, 6, 9, 1, 3, 4, 1, 3, 6, 0, 4, 4, 9, 7, 3, 2, 2, 5, 6, 6, 4, 4, 7, 5, 2, 5, 7, 8, 4, 8, 8, 9, 8, 1, 0, 8, 1, 8, 1, 4, 5, 7, 1, 4, 7, 9, 7, 1, 2, 5, 7, 4, 8, 0

Offset: 0

Dumitru Damian, Oct 31 2021

			-0.15766014916783233039054467406996221822374946546295676913413604497322566...

Cf. A072691, A086722, A086724, A111003, A222171.

RealDigits[Integrate[x*Log[x]/(1 + x + x^2), {x, 0, 1}], 10, 100][[1]] (* Amiram Eldar, Oct 31 2021 *)
RealDigits[Pi^2/54 - PolyGamma[1, 2/3]/9, 10, 100][[1]] (* Vaclav Kotesovec, Oct 31 2021 *)

intnum(x=0, 1, x*log(x)/(1+x+x^2)) \\ Michel Marcus, Oct 31 2021

RealField(25)(numerical_integral(x*log(x)/(1+x+x^2), 0, 1)[0])

Equals Pi^2/54 - PolyGamma(1, 2/3)/9. - Vaclav Kotesovec, Oct 31 2021

cp's OEIS Frontend

A113262 One quarter of the number of solutions to a^2 + b^2 + 3*c^2 + 3*d^2 = n.

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A185152 Expansion of (q/2) * phi(q)^3 (d/dq) phi(q) in powers of q.

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A369180 Alternating sum of the k-adic valuations (ruler functions) of n.

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A276712 Decimal expansion of zeta(3)/8.

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A228274 a(n) = Sum_{d|n, n/d odd} n * d.

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A282468 Decimal expansion of the zeta function at 2 of every second prime number.

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A343481 a(n) is the sum of all digits of n in every prime base 2 <= p <= n.

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A113262 One quarter of the number of solutions to a^2 + b^2 + 3c^2 + 3d^2 = n.