A086463 -id:A086463 - cp's OEIS frontend

A261852 Decimal expansion of the central binomial sum S(8), where S(k) = Sum_{n>=1} 1/(n^k binomial(2n,n)).

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

Offset: 0

Jean-François Alcover, Sep 03 2015

			0.5006588912976705433145571270829868383840732523404540388886438...

J. M. Borwein, D. J. Broadhurst, J. Kamnitzer, Central Binomial Sums, Multiple Clausen Values and Zeta Values, arXiv:hep-th/0004153, 2000.
Eric Weisstein's MathWorld, Central Binomial Coefficient

Cf. A073010 (S(1)), A086463 (S(2)), A145438 (S(3)), A086464 (S(4)), A261839 (S(5)), A261850 (S(6)), A261851 (S(7)).

S[8] = Sum[1/(n^8*Binomial[2n, n]), {n, 1, Infinity}]; RealDigits[S[8], 10, 100] // First

Equals (1/2) 8F7(1,...,1; 3/2,2,...,2; 1/4).

Also equals (4/45)*Integral_{0..Pi/3} t*log(2*sin(t/2))^6 dt.

A291900 Sum of the divisors of 24*n - 1, divided by 24, minus n.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 0, 0, 2, 0, 0, 2, 0, 3, 0, 0, 2, 0, 9, 0, 0, 2, 2, 7, 0, 4, 0, 3, 6, 0, 0, 3, 5, 7, 0, 0, 0, 0, 15, 6, 0, 3, 0, 9, 4, 0, 10, 0, 13, 5, 0, 3, 3, 22, 0, 4, 0, 5, 12, 0, 19, 0, 0, 13, 0, 0, 0, 10, 14, 4, 6, 7, 5, 19, 11, 0, 0, 0, 16, 5, 4, 12, 8, 28, 0, 0, 0, 0, 35, 6, 4, 0, 5, 32, 4, 18, 8, 0, 31, 0

Offset: 1

Omar E. Pol, Nov 02 2017

The indices of the zeros give A131210.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000203, A008606, A086463, A131210, A134517, A183010, A183011, A280097, A280098, A294614.

a[n_] := DivisorSigma[1, 24 n - 1]/24 - n; Array[a, 90] (* Robert G. Wilson v, Nov 04 2017 *)

a(n) = sigma(24*n-1)/24 - n; \\ Michel Marcus, Nov 04 2017

a(n) = sigma(24*n-1)/24 - n = A000203(A183010(n))/24 - n = A280097(n)/24 - n = A280098(n) - n.

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/18 - 1/2 = 0.048311... . - Amiram Eldar, Mar 28 2024

A062785 a(n) = Chowla's function of n * sigma(n).

Original entry on oeis.org

0, 0, 0, 14, 0, 60, 0, 90, 39, 126, 0, 420, 0, 216, 192, 434, 0, 780, 0, 882, 320, 468, 0, 2100, 155, 630, 480, 1512, 0, 2952, 0, 1890, 672, 1026, 576, 4914, 0, 1260, 896, 4410, 0, 5088, 0, 3276, 2496, 1800, 0, 9300, 399, 3906, 1440, 4410, 0, 7800, 1152, 7560, 1760, 2790, 0, 17976, 0, 3168, 4160, 7874, 1512

Offset: 1

Jason Earls, Jul 18 2001

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000203, A048050.

Cf. A002117, A086463.

a[n_] := Module[{s = DivisorSigma[1, n]}, s*(s - n - 1)]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Apr 01 2024 *)

a(n) = {my(s = sigma(n)); if(n == 1, 0, s*(s-n-1));}

a(n) = A000203(n)*A048050(n). - Michel Marcus, Jun 29 2018

Sum_{k=1..n} a(k) ~ (5*zeta(3)/6 - Pi^2/18) * n^3. - Amiram Eldar, Apr 01 2024

a(1) corrected by Amiram Eldar, Apr 01 2024

A294614 Sum of the divisors of 12*n - 1, divided by 12, minus n.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 0, 2, 3, 0, 0, 0, 3, 4, 0, 0, 0, 0, 8, 4, 3, 0, 3, 6, 0, 0, 5, 0, 7, 4, 0, 0, 0, 18, 0, 0, 0, 0, 9, 4, 12, 4, 0, 14, 0, 0, 5, 8, 11, 0, 0, 6, 0, 12, 9, 0, 5, 0, 13, 6, 5, 10, 7, 14, 0, 0, 5, 0, 31, 0, 5, 0, 7, 30, 0, 12, 0, 0, 17, 6, 0, 0, 13, 18, 9, 8

Offset: 1

Omar E. Pol and Robert G. Wilson v, Nov 04 2017

a(n) = 0 iff n is in A138620.

First occurrence of k > -1: 1, 3, 8, 13, 18, 31, 28, 33, 23, 43, 66, 53, 45, 63, 48, 101, 166, etc.

			a(13) = 3 since d(12*13-1)/12 - 13 = 192/12 - 13 = 16 - 13 = 3.

Inspired by A291900.

Cf. A000203, A017653, A068231, A086463, A138620.

a[n_] := DivisorSigma[1, 12 n - 1]/12 - n; Array[a, 90]

PARI
```
a(n) = sigma(12*n-1)/12 - n;
```

a(n) = sigma(12*n-1)/12 - n = A000203(A017653(n-1))/12 - n.

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/18 - 1/2 = 0.048311... . - Amiram Eldar, Mar 28 2024

A295012 a(n) = sigma(12n - 1)/12, where sigma = sum of divisors (A000203).

Original entry on oeis.org

1, 2, 4, 4, 5, 6, 7, 10, 9, 12, 11, 14, 16, 14, 15, 16, 20, 22, 19, 20, 21, 22, 31, 28, 28, 26, 30, 34, 29, 30, 36, 32, 40, 38, 35, 36, 37, 56, 39, 40, 41, 42, 52, 48, 57, 50, 47, 62, 49, 50, 56, 60, 64, 54, 55, 62, 57, 70, 68, 60, 66, 62, 76, 70, 70, 76

Offset: 1

M. F. Hasler, Dec 08 2017

Robert G. Wilson v observes in A280098 that {1, 3, 4, 6, 8, 12, 24} seem to be the only positive integers k such that sigma(kn-1)/k is an integer for all n > 0.

Muniru A Asiru, Table of n, a(n) for n = 1..100000

Cf. A280098 (analog for k = 24), A097723 (analog for k = 4), A033686 (analog for k = 3), A000203 (sigma, also the  analog for k = 1).
The analog for k = 8 is A258835, up to the offset.
The analog for k = 6 is A098098 (up to the offset), a signed variant of this and the preceding one is A258831.
Cf. A086463.

sequence := List([1..10^5], n-> Sigma(12 *n-1)/12); # Muniru A Asiru, Dec 28 2017

with(numtheory):
seq(sigma(12*n-1)/12, n=1..10^3); # Muniru A Asiru, Dec 28 2017

Array[DivisorSigma[1, 12 # - 1]/12 &, 66] (* Michael De Vlieger, Dec 08 2017 *)

PARI
```
vector(90,n,sigma(12*n-1)/12)
```

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/18 = 0.548311... (A086463). - Amiram Eldar, Mar 28 2024

A306198 Multiplicative with a(p^e) = p^(e-1)*(p^2 - p - 1).

Original entry on oeis.org

1, 1, 5, 2, 19, 5, 41, 4, 15, 19, 109, 10, 155, 41, 95, 8, 271, 15, 341, 38, 205, 109, 505, 20, 95, 155, 45, 82, 811, 95, 929, 16, 545, 271, 779, 30, 1331, 341, 775, 76, 1639, 205, 1805, 218, 285, 505, 2161, 40, 287, 95, 1355, 310, 2755, 45, 2071, 164, 1705, 811

Offset: 1

Jianing Song, Jan 28 2019

For any positive integer n and any m coprime to n, define R(n,m) = Product_{primes p divides n} (p - [m == 1 (mod p)]), where [] is an Iverson branket. Then we have the following conjecture: (Start)
Let k == 2, 3 (mod 4) be a squarefree number, b be any positive integer such that k*b^2 is not a perfect power and not equal to -1, n be either coprime to or divisible by 4*k. Define Q(N,k*b^2,n,m) = # {primes p <= N : p == m (mod n), k*b^2 is a primitive modulo p}, then:
(a) if gcd(n, 4*k) = 1, then Q(N,k*b^2,n,m)/(C*PrimePi(N)) ~ R(n,m)/a(n);
(b) if 4*k divides n, then Q(N,k*b^2,n,m)/(C*PrimePi(N)) ~ 2*R(n,m)/a(n) if Jacobi(k/m) = -1 and 0 if Jacobi(k/m) = +1,
Where C is the Artin's constant = A005596, PrimePi = A000720. (End)
(Note that Sum_{m=1..n, gcd(m,n)=1} R(n,m) = a(n).)
For example, let N = 10^6:
k*b^2 |  n |  m | Q(N,k*b^2,n,m) | Q(N,k*b^2,n,m)/(C*PrimePi(N))
   2  |  8 |  3 |      14642     |   0.498794... approx = 2/4
   3  |  5 |  1 |       6192     |   0.210936... approx = 4/19
  -2  | 48 | 13 |       2933     |   0.099915... approx = 4/40
  -5  |  9 |  5 |       5933     |   0.202113... approx = 3/15

Jianing Song, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Artin's constant.
Wikipedia, Artin's conjecture on primitive roots.

Cf. A000720 (PrimePi), A005596 (Artin's constant), A086463.

P := (p, e) -> p^(e-1)*(p^2 - p - 1):
a := n -> mul(P(f[1], f[2]), f in ifactors(n)[2]):
seq(a(n), n=1..58); # Peter Luschny, Feb 13 2019

a[n_] := Product[{p, e} = pe; p^(e-1) (p^2-p-1), {pe, FactorInteger[n]}]; a[1] = 1; Array[a, 58] (* Jean-François Alcover, Jul 22 2019 *)

a(n) = my(f=factor(n)); prod(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]); (p^2 - p - 1)*p^(e-1))

Sum_{k=1..n} a(k) ~ c * n^3, where c = (Pi^2/18) * Product_{p prime} (1 - 3/p^2 + 1/p^3 + 1/p^4) = 0.1314639252... . - Amiram Eldar, Dec 01 2022

A355058 Numbers m such that d(m) mod 6 = 3, where d(m) is the number of divisors of m.

Original entry on oeis.org

4, 9, 25, 36, 49, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 676, 784, 841, 900, 961, 1089, 1156, 1225, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2500, 2601, 2704, 2809, 2916, 3025, 3136, 3249, 3364

Offset: 1

Michael De Vlieger, Jul 04 2022

All terms are square; contains squares of primes.

			Let p be a prime; p^2 has 3 divisors {1, p, p^2}, therefore all squares of primes {4, 9, 25, 49, ...} are in the sequence.
36 is in the sequence because d(36) = 9, and 9 mod 6 = 3.
16 is not in the sequence because it has 5 divisors, and 5 mod 6 = 5.

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

Cf. A000005, A000290, A001248, A086463, A352475.

Select[Range[2^12], Mod[DivisorSigma[0, #], 6] == 3 &]

isok(m) = (numdiv(m) % 6) == 3; \\ Michel Marcus, Jul 05 2022

from itertools import count, islice
from sympy import factorint, prod
def A355058_gen(): # generator of terms
    return map(lambda n:n**2,filter(lambda n:prod((2*e+1)%6 for e in factorint(n).values())%6==3,count(1)))
A355058_list = list(islice(A355058_gen(),30)) # Chai Wah Wu, Jul 06 2022

{ A000290 \ A352475 }.

Sum_{n>=1} 1/a(n) = Pi^2/18 (A086463). - Amiram Eldar, Jul 06 2022

A246686 Decimal expansion of 'mu', a percolation constant associated with the asymptotic threshold for 3-dimensional bootstrap percolation.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 01 2014

			0.4039127202987558379321142074495349887102719293775432644...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.18 Percolation Cluster Density Constants, pp. 371-378.

J. Balogh, B. Bollobás and R. Morris, Bootstrap percolation in three dimensions. arXiv:0806.4485 [math.CO], 2008-2009.
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2024; p. 47.
Eric Weisstein's MathWorld, Bootstrap Percolation

Cf. A086463 (analog 2-dimensional percolation constant).

mu = -NIntegrate[Log[1/2 - Exp[-2*x]/2 + (1/2)*Sqrt[1 + Exp[-4*x] - 4*Exp[-3*x] + 2 *Exp[-2*x]]] , {x, 0, Infinity}, WorkingPrecision -> 101]; RealDigits[mu] // First

Equals -Integral_{0..oo} log(1/2 - exp(-2*x)/2 + (1/2)*sqrt(1 + exp(-4*x) - 4*exp(-3*x) + 2*exp(-2*x))) dx.

A321675 a(n) = Sum_{k=1..10^n} k*sigma(k).

Original entry on oeis.org

1, 622, 558275, 549175530, 548429473046, 548320905633448, 548312690631798482, 548311465139943768941, 548311366911386862908968, 548311356554322895313137239, 548311355740964925044531454428, 548311355626818302486560961291870, 548311355617569600726982364186141942

Offset: 0

Daniel Suteu, Nov 16 2018

nonn

Cf. A000203, A064987, A072692, A086463 (Pi^2/18), A143128.

Array[Sum[k DivisorSigma[1, k], {k, 10^#}] &, 7, 0] (* Michael De Vlieger, Nov 20 2018 *)

a(n) = sum(k=1, 10^n, k*sigma(k)); \\ Michel Marcus, Nov 23 2018

a(n) = A143128(10^n).

a(n) ~ 10^(3*n) * Pi^2 / 18.

cp's OEIS Frontend

A261852 Decimal expansion of the central binomial sum S(8), where S(k) = Sum_{n>=1} 1/(n^k binomial(2n,n)).

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A291900 Sum of the divisors of 24*n - 1, divided by 24, minus n.

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A062785 a(n) = Chowla's function of n * sigma(n).

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A294614 Sum of the divisors of 12*n - 1, divided by 12, minus n.

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A295012 a(n) = sigma(12n - 1)/12, where sigma = sum of divisors (A000203).

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A306198 Multiplicative with a(p^e) = p^(e-1)*(p^2 - p - 1).

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A355058 Numbers m such that d(m) mod 6 = 3, where d(m) is the number of divisors of m.

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