A065489 -id:A065489 - cp's OEIS frontend

A065463 Decimal expansion of Product_{p prime} (1 - 1/(p*(p+1))).

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

Offset: 0

N. J. A. Sloane, Nov 19 2001

The density of A268335. - Vladimir Shevelev, Feb 01 2016

The probability that two numbers are coprime given that one of them is coprime to a randomly chosen third number. - Luke Palmer, Apr 27 2019

			0.7044422009991655927366033503...

Olivier Bordellès and Benoit Cloitre, An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions, J. Int. Seq., Vol. 16 (2013), Article 13.6.3.
Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Mathematische Zeitschrift, Vol. 74, No. 1 (1960), pp. 66-80.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 50.
David Handelman, Invariants for critical dimension groups and permutation-Hermite equivalence, arXiv preprint arXiv:1309.7417 [math.AC], 2013-2017.
R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arxiv:0903.2514 [math.NT] (2009) constant Q_1^(1).
G. Niklasch, Some number theoretical constants: 1000-digit values.
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocals of some multiplicative functions, Mathematical Journal of Okayama University, Vol. 21, No. 2 (1979), pp. 155-164.
R. Sitaramachandrarao and D. Suryanarayana, On Sigma_{n<=x} sigma*(n) and Sigma_{n<=x} phi*(n), Proceedings of the American Mathematical Society, Vol. 41, No. 1 (1973), pp. 61-66.
László Tóth, Alternating sums concerning multiplicative arithmetic functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1,arXiv preprint, arXiv:1608.00795 [math.NT], 2016.
Deyu Zhang and Wenguang Zhai, Mean Values of a Gcd-Sum Function Over Regular Integers Modulo n, J. Int. Seq., Vol. 13 (2010), Article 10.4.7, eq. (4).
Rimer Zurita Generalized Alternating Sums of Multiplicative Arithmetic Functions, J. Int. Seq., Vol. 23 (2020), Article 20.10.4.

Cf. A001615, A007947, A047994, A078082, A268335, A306633.

Cf. A000203, A065473, A065480, A065490, A008683.

$MaxExtraPrecision = 1200; digits = 98; terms = 1200; P[n_] := PrimeZetaP[n]; LR = Join[{0, 0}, LinearRecurrence[{-2, 0, 1}, {-2, 3, -6}, terms + 10]]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*P[n - 1]/(n - 1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 18 2016 *)

prodeulerrat(1 - 1/(p*(p+1))) \\ Amiram Eldar, Mar 14 2021

From Amiram Eldar, Mar 05 2019: (Start)
Equals lim_{m->oo} (2/m^2)*Sum_{k=1..m} rad(k), where rad(k) = A007947(k) is the squarefree kernel of k (Cohen).
Equals lim_{m->oo} (2/m^2)*Sum_{k=1..m} uphi(k), where uphi(k) = A047994(k) is the unitary totient function (Sitaramachandrarao and Suryanarayana).
Equals lim_{m->oo} (1/log(m))*Sum_{k=1..m} 1/psi(k), where psi(k) = A001615(k) is the Dedekind psi function (Sita Ramaiah and Suryanarayana).
(End)
Equals A065473*A013661/A065480. - Luke Palmer, Apr 27 2019
Equals Sum_{k>=1} mu(k)/(k*sigma(k)), where mu is the Möbius function (A008683) and sigma(k) is the sum of divisors of k (A000203). - Amiram Eldar, Jan 14 2022
Equals 1/A065489. - R. J. Mathar, May 27 2025

A167338 Totally multiplicative sequence with a(p) = p*(p+1) = p^2+p for prime p.

Original entry on oeis.org

1, 6, 12, 36, 30, 72, 56, 216, 144, 180, 132, 432, 182, 336, 360, 1296, 306, 864, 380, 1080, 672, 792, 552, 2592, 900, 1092, 1728, 2016, 870, 2160, 992, 7776, 1584, 1836, 1680, 5184, 1406, 2280, 2184, 6480, 1722, 4032, 1892, 4752, 4320, 3312, 2256, 15552

Offset: 1

Jaroslav Krizek, Nov 01 2009

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

Cf. A003959, A065489, A133205.

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 1)^fi[[All, 2]])); Table[a[n]*n, {n, 1, 100}] (* G. C. Greubel, Jun 06 2016 *)

for(n=1, 100, print1(direuler(p=2, n, (1 + 1/(1/X/p - p - 1))/(1 - p^2*X))[n], ", ")) \\ Vaclav Kotesovec, Apr 05 2023

Multiplicative with a(p^e) = (p*(p+1))^e.
If n = Product p(k)^e(k) then a(n) = Product (p(k)*(p(k)+1))^e(k).
a(n) = n * A003959(n).
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/(p^2 + p - 1)) = A065489 = 1.419562880505485919317235861789735359166071586305122542698983695564330971... - Vaclav Kotesovec, Sep 20 2020
Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/Pi^2) / Product_{p prime} (1 - 2/p^2 - 1/p^3) = 0.8913709085... . - Amiram Eldar, Dec 15 2022, c = A065488/3. - Vaclav Kotesovec, Apr 05 2023
Dirichlet g.f.: zeta(s-2) * Product_{p prime} (1 + 1/(p^(s-1) - p - 1)). - Vaclav Kotesovec, Apr 05 2023

A078077 Continued fraction expansion of Product_{p prime} (1 + 1/(p^2+p-1)).

Original entry on oeis.org

1, 2, 2, 1, 1, 1, 1, 4, 2, 1, 1, 3, 703, 2, 1, 1, 1, 3, 5, 1, 1, 4, 10, 2, 10, 2, 1, 2, 1, 4, 4, 11, 5, 1, 2, 1, 1, 30, 37, 2, 75, 1, 1, 8, 5, 1, 1, 7, 44, 280, 5, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 3, 1, 9, 94, 1, 127, 2, 2, 7, 3, 3, 3, 6, 2, 1, 705, 24, 2, 7, 2, 2, 1, 3, 7, 1, 1, 3, 40, 6, 1

Offset: 0

Benoit Cloitre, Dec 02 2002

Cf. A065489 (decimal expansion).

contfrac(prodeulerrat(1 + 1/(p^2+p-1))) \\ Amiram Eldar, Mar 15 2021

Offset changed by Andrew Howroyd, Jul 05 2024

A319597 Number of conjugacy classes for a non-abelian group of order p^3, where p is prime: a(n) = p^2 + p - 1 where p = prime(n).

Original entry on oeis.org

5, 11, 29, 55, 131, 181, 305, 379, 551, 869, 991, 1405, 1721, 1891, 2255, 2861, 3539, 3781, 4555, 5111, 5401, 6319, 6971, 8009, 9505, 10301, 10711, 11555, 11989, 12881, 16255, 17291, 18905, 19459, 22349, 22951, 24805, 26731, 28055, 30101, 32219, 32941, 36671

Offset: 1

Juan Lanfranco, Sep 23 2018

nonn

For a non-abelian group of order p^3, we can use the class equation, p-group has nontrivial center result, group modulo center is cyclic implies group is abelian result, and the orbit-stabilizer theorem to give the number of conjugacy classes and number of elements in each conjugacy class.

The elements of A028387 with prime index.

			For p^3=2^3=8, the conjugacy classes of the Dihedral group = <r, s | r^4=1, s^2=1, srs=r^{-1}> are {1}, {r^2}, {r, r^3}, {s, sr^2}, {sr, sr^3}.

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

Cf. A000040, A028387, A065480, A065489.

A028387:= n -> n^2+n-1:
seq(A028387(ithprime(i)),i=1..50); # Robert Israel, Dec 23 2018

f[n_]:=n^2 + n - 1 ; f[Prime[Range[43]]] (* Amiram Eldar, Nov 21 2018 *)

a(n) = {my(p = prime(n)); p^2 + p - 1; } \\ Amiram Eldar, Nov 07 2022

From Amiram Eldar, Nov 07 2022: (Start)
a(n) = A028387(A000040(n)-1).
Product_{n>=1} (1 + 1/a(n)) = A065489.
Product_{n>=1} (1 - 1/a(n)) = A065480. (End)

A382664 Partial sums of the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 3, 6, 11, 17, 24, 32, 42, 53, 66, 80, 95, 112, 131, 152, 174, 197, 221, 247, 274, 303, 333, 364, 396, 429, 463, 498, 535, 573, 612, 652, 693, 735, 778, 824, 871, 922, 975, 1029, 1084, 1140, 1197, 1255, 1314, 1375, 1437, 1502, 1568, 1635, 1704, 1774, 1845, 1918

Offset: 1

Amiram Eldar, Apr 02 2025

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

Cf. A065489, A268335.

Similar sequences: A173143, A174172, A358038, A362971.

Accumulate[Select[Range[100], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &]]

isexpodd(n) = {my(f = factor(n)); for(i=1, #f~, if(!(f[i, 2] % 2), return (0))); 1;}
list(lim) = {my(s = 0); for(k = 1, lim, if(isexpodd(k), s += k; print1(s, ", "))); }

a(n) = Sum_{k=1..n} A268335(k).

a(n) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p^2+p-1)) = 1.419562... (A065489).

A340565 Decimal expansion of the Product_{lesser twin primes p == 5 (mod 6)} 1/(1 - 1/p^2).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jan 11 2021

Lesser twin primes A001359 (with the exception of the first prime, 3) are congruent to 5 mod 6: this constant is smaller than A340576.
By extrapolating method most probably the next two decimal digits are 1.056932291(46).
The known high-precision algorithms for Euler products are based on the Dirichlet L function and the Moebius inversion formula (see Mathematica procedure of Jean-François Alcover in A175646).
The constant is between 1.056932291453... and 1.056932291494. - R. J. Mathar, Feb 14 2025

			1.0569322914...

Cf. A005596, A065463, A065465, A065483, A065485, A065489, A065493, A072691, A082695, A111003, A175646.

One more digit confirmed by a bracketing of partial products - R. J. Mathar, Feb 14 2025

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A065463 Decimal expansion of Product_{p prime} (1 - 1/(p*(p+1))).

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A167338 Totally multiplicative sequence with a(p) = p*(p+1) = p^2+p for prime p.

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A078077 Continued fraction expansion of Product_{p prime} (1 + 1/(p^2+p-1)).

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A319597 Number of conjugacy classes for a non-abelian group of order p^3, where p is prime: a(n) = p^2 + p - 1 where p = prime(n).

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A382664 Partial sums of the exponentially odd numbers (A268335).

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A340565 Decimal expansion of the Product_{lesser twin primes p == 5 (mod 6)} 1/(1 - 1/p^2).

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