A080130 -id:A080130 - cp's OEIS frontend

A218342 Decimal expansion of e^-gamma * Product_(1 - 1/(p^3 - p^2 - p + 1)) where the product is over all primes p.

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

Offset: 0

Charles R Greathouse IV, Oct 26 2012

The average order of Carmichael's lambda function is x/log x * exp(B log log x/log log log x (1 + o(1))), where B is this constant. Under the GRH, the same applies to A036391(n)/n, the sum of the orders mod n of the numbers coprime to n divided by n.

			0.34537206410298648767349682789103371072066562538041...

Paul Erdős, Carl Pomerance, and Eric Schmutz, Carmichael's lambda function, Acta Arithmetica 58 (1991), pp. 363-385.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 156 (constant C9).
Sungjin Kim, On the order of 'a' modulo 'n' on average, International Journal of Number Theory, Vol. 12, No. 8 (2016), pp. 2073-2080; arXiv preprint, arXiv:1509.03768 [math.NT], 2015-2016.
Pär Kurlberg and Carl Pomerance, On a problem of Arnold: the average multiplicative order of a given integer, Algebra & Number Theory, Vol. 7, No. 4 (2013), pp. 981-999; arXiv preprint, arXiv:1108.5209 [math.NT], 2012.
R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], 2009-2011; Eq. (106) page 17.

Cf. A002322, A036391, A080130, A162578.

$MaxExtraPrecision = 200; m0 = 1000; dm = 200; digits = 105; Clear[f]; f[m_] := f[m] = (slog = Normal[Series[Log[1 - 1/((p - 1)^2*(p + 1))], {p, Infinity, m}]]; Exp[slog] /. Power[p, n_] -> PrimeZetaP[-n] // N[#, digits + 10] &); f[m = m0]; Print[m, " ", f[m]]; f[m = m + dm]; While[Print[m, " ", f[m]]; RealDigits[f[m], 10, digits + 5] !=  RealDigits[f[m - dm], 10, digits + 5], m = m + dm]; B = Exp[-EulerGamma]*f[m]; RealDigits[B, 10, digits] // First (* Jean-François Alcover, Sep 20 2015 *)

exp(-Euler) * prodeulerrat(1-1/((p-1)^2*(p+1))) \\ Amiram Eldar, Mar 09 2021

More digits from Jean-François Alcover, Sep 20 2015

A335636 Expansion of e.g.f. Product_{k>0} 1/(1 - tan(x)^k / k).

Original entry on oeis.org

1, 1, 3, 13, 80, 560, 4972, 48060, 552632, 6813560, 95846728, 1435488184, 23855755040, 419889384096, 8048166402304, 162616435301824, 3531256457687168, 80497793591765120, 1953028123616286592, 49561115477458450560, 1328614915154244276224, 37134707962379971432448

Offset: 0

Seiichi Manyama, Oct 03 2020

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..430

Cf. A000182, A007841, A335627, A335635, A335638, A335643.

max = 21; Range[0, max]! * CoefficientList[Series[Product[1/(1 - Tan[x]^k/k), {k, 1, max}], {x, 0, max}], x] (* Amiram Eldar, Oct 03 2020 *)

N=40; x='x+O('x^N); Vec(serlaplace(1/prod(k=1, N, 1-tan(x)^k/k)))

N=40; x='x+O('x^N); Vec(serlaplace(exp(sum(i=1, N, sum(j=1, N\i, tan(x)^(i*j)/(i*j^i))))))

E.g.f.: exp( Sum_{i>0} Sum_{j>0} tan(x)^(i*j)/(i*j^i) ).

Conjecture: a(n) ~ A080130 * n * 2^(2*n+2) * n! / Pi^(n+2). - Vaclav Kotesovec, Oct 04 2020

A335638 Expansion of e.g.f. Product_{k>0} (1 + tan(x)^k / k).

Original entry on oeis.org

1, 1, 1, 7, 22, 190, 1170, 11646, 109520, 1289168, 16018064, 223757840, 3407971488, 55709905056, 998011344928, 18778681069024, 385316251841536, 8225863823985664, 189755182485906432, 4538893733746003968, 116147781156885837824, 3078530007519830730752, 86521073899573883088896

Offset: 0

Seiichi Manyama, Oct 03 2020

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..430

Cf. A000182, A007838, A335630, A335636, A335637, A336046.

max = 22; Range[0, max]! * CoefficientList[Series[Product[1 + Tan[x]^k/k, {k, 1, max}], {x, 0, max}], x] (* Amiram Eldar, Oct 03 2020 *)

N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, 1+tan(x)^k/k)))

N=40; x='x+O('x^N); Vec(serlaplace(exp(sum(i=1, N, sum(j=1, N\i, (-1)^(i+1)*tan(x)^(i*j)/(i*j^i))))))

E.g.f.: exp( Sum_{i>0} Sum_{j>0} (-1)^(i+1)*tan(x)^(i*j)/(i*j^i) ).

Conjecture: a(n) ~ A080130 * 2^(2*n+1) * n! / Pi^(n+1). - Vaclav Kotesovec, Oct 04 2020

A368246 Number of permutations of [n] whose cycle minima sum to n.

Original entry on oeis.org

1, 1, 0, 2, 3, 8, 90, 384, 2940, 18864, 232848, 1919520, 23364000, 261282240, 3486637440, 48900116160, 746747164800, 11559784320000, 201817271416320, 3580457619916800, 68121866659875840, 1366946563510886400, 28802183294533017600, 627950275273991577600

Offset: 0

Alois P. Heinz, Dec 18 2023

nonn

Also the number of permutations of [n] for which the sum of the positions of the left-to-right maxima is n: a(4) = 3: 2143, 3142, 3241; a(5) = 8: 31254, 32154, 41253, 41352, 42153, 42351, 43152, 43251.

			a(0) = 1: the empty permutation.
a(1) = 1: (1).
a(2) = 0.
a(3) = 2: (13)(2), (1)(23).
a(4) = 3: (124)(3), (142)(3), (12)(34).
a(5) = 8: (1235)(4), (1253)(4), (1325)(4), (1352)(4), (1523)(4), (1532)(4), (123)(45), (132)(45).

Alois P. Heinz, Table of n, a(n) for n = 0..451
Wikipedia, Permutation

Main diagonal of A143946.

Cf. A001620, A080130, A368204, A368678.

b:= proc(n) option remember; `if`(n=0, 1,
      expand(b(n-1)*(x^n+n-1)))
    end:
a:= n-> coeff(b(n), x, n):
seq(a(n), n=0..23);

a(n) = A143946(n,n).

a(n) ~ c * (n-1)!, where c = 0.561459..., conjecture: c = exp(-gamma) = A080130, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Dec 29 2023

A242909 Decimal expansion of exp(-gamma/2).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 26 2014

			0.7493060012884490236058715186852615118333...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 3.10 Kneser-Mahler polynomial constants p. 234.

C. J. Smyth, On measures of polynomials in several variables, Bulletin of the Australian Mathematical Society, Volume 23 (1981), Issue 01.

Cf. A073004, A080130, A242908, A242910.

Exp(-EulerGamma(100)/2); // Stefano Spezia, Dec 10 2024

RealDigits[Exp[-EulerGamma/2], 10, 101] // First

Lim_(m->oo) M(z_1, z_2, ..., z_m)/sqrt(m), where M is Mahler's measure for multivariate polynomials.

A244274 Decimal expansion of e*gamma, the product of Euler number and Euler-Mascheroni constant.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Jun 28 2014

			1.569034853003742285079907849123151192307242907588894908656654...

Stanislav Sykora, Table of n, a(n) for n = 1..2000

Cf. A001113, A001620, A073004, A080130, A244499.

R:= RealField(100); Exp(1)*EulerGamma(R); // G. C. Greubel, Aug 30 2018

RealDigits[E*EulerGamma,10,120][[1]] (* Harvey P. Dale, Jan 01 2015 *)

PARI
```
exp(1)*Euler
```

A246499 Decimal expansion of zeta(2)/exp(gamma), gamma being the Euler-Mascheroni constant.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Nov 14 2014

It follows from Mertens theorem that this constant is the limit for large m of log(prime(m))*Product_{k=1..m} 1/(1 + 1/prime(k)).

			0.9235638316741813823235099539877039168469319632611163252035958316...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Eric Weisstein's World of Mathematics, Mertens Theorem, Equations 5-9

Cf. A001113, A000796, A001620, A013661, A073004, A080130, A097663.

R:=RealField(100); Pi(R)^2/(6*Exp(EulerGamma(R))); // G. C. Greubel, Aug 30 2018

RealDigits[Zeta[2]/E^EulerGamma, 10, 100][[1]] (* Alonso del Arte, Nov 14 2014 *)

PARI
```
Pi^2/6/exp(Euler)
```

Equals Pi^2/(6*exp(gamma)).
Equals lim_{m->infinity} log(prime(m))*Product_{k=1..m} 1/(1 + 1/prime(k)).
Equals A013661/A073004. - Michel Marcus, Nov 18 2014

A370689 Numerator of sigma(phi(n))/phi(sigma(n)), where sigma is the sum of the divisors function and phi is the Euler totient function.

Original entry on oeis.org

1, 1, 3, 1, 7, 3, 3, 7, 1, 7, 9, 7, 14, 3, 15, 1, 31, 1, 39, 5, 7, 3, 9, 15, 7, 7, 39, 7, 7, 5, 9, 31, 21, 31, 15, 7, 91, 39, 5, 31, 15, 7, 24, 7, 5, 3, 9, 31, 8, 7, 21, 10, 49, 39, 15, 15, 91, 7, 45, 31, 28, 9, 91, 1, 31, 7, 9, 7, 21, 5, 6, 5, 65, 91, 3, 91, 21

Offset: 1

Amiram Eldar, Feb 27 2024

			Fractions begin with: 1, 1/2, 3/2, 1/2, 7/2, 3/4, 3, 7/8, 1, 7/6, 9/2, 7/12, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck and Florian Luca, On the composition of the Euler function and the sum of divisors function, Colloquium Mathematicum, Vol. 108, No. 1 (2007), pp. 31-51.

Cf. A000010, A000203, A033632, A062401, A062402, A065395, A066930, A289336, A073858 (positions of 1's), A289412, A370690 (denominators).

Cf. A080130, A091724.

Table[DivisorSigma[1, EulerPhi[n]]/EulerPhi[DivisorSigma[1, n]], {n, 1, 100}] // Numerator

a(n) = {my(f = factor(n)); numerator(sigma(eulerphi(f)) / eulerphi(sigma(f)));}

Let f(n) = a(n)/A370690(n) = A062402(n)/A062401(n).
Formulas from De Koninck and Luca (2007):
lim sup_{n->oo} f(n)/log_2(n)^2 = exp(2*gamma) (A091724).
lim inf_{n->oo} f(n)/log_2(n)^2 = delta exists, and exp(-gamma)/40 <= delta <= 2*exp(-gamma).
Sum_{k=1..n} f(k) = c * exp(2*gamma) * log_3(n)^2 * n + O(n * log_3(n)^(3/2)), where c = Product_{p prime} (1 - 3/(p*(p + 1)) + 1/(p^2*(p + 1)) + ((p-1)^3/p^2)*Sum_{k>=3} 1/(p^k-1)) = 0.45782563109026414241... .

A061091 Number of k with 1 <= k <= n relatively prime to phi(k).

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 21, 21, 21, 21, 21, 21, 22, 22, 23, 23, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 28

Offset: 1

Frank Ellermann, May 29 2001

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 115-119.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Joel E. Cohen, Conjectures about Primes and Cyclic Numbers, arXiv:2508.08335 [math.NT], 2025. See p. 2.
Paul Erdős, Some asymptotic formulas in number theory, J. Indian Math. Soc. 12 (1948) 75-78.
Steven R. Finch, Euler Totient Function Asymptotic Constants. [Broken link]
Steven R. Finch, Euler Totient Function Asymptotic Constants. [From the Wayback machine]
Paul Pollack, Numbers which are orders only of cyclic groups, Proceedings of the American Mathematical Society, Vol. 150, No. 2 (2022), pp. 515-524; arXiv preprint, arXiv:2007.09734 [math.NT], 2020.
Imre Z. Ruzsa, Erdős and the integers, J. Number Theory, Vol. 79, No. 1 (1999), 115-163.

Partial sums of A297086.

Cf. A000010 (phi), A001620 (gamma), A003277, A073004, A080130.

s[n_] := Boole[CoprimeQ[n, EulerPhi[n]]]; Accumulate[Array[s, 100]]  (* Amiram Eldar, Dec 10 2024 *)

a(n) = sum(k=1, n, gcd(k, eulerphi(k)) == 1) \\ Charles R Greathouse IV, Jan 29 2013 (corrected by Iain Fox, Dec 25 2017)

list(lim) = {my(s = 0); for(k = 1, lim, s += gcd(k, eulerphi(k)) == 1; print1(s, ", "));} \\ Amiram Eldar, Dec 10 2024

Limit_{n->oo} a(n) * log(log(log(n))) / n = 1/exp(gamma).
a(n) = Sum_{k=1..n} gcd(k, phi(k)) = 1.
a(1) = 1; a(n) = a(n-1) + A297086(n). - Iain Fox, Dec 25 2017

A215640 Sum of divisors of colossally abundant numbers.

Original entry on oeis.org

3, 12, 28, 168, 360, 1170, 9360, 19344, 232128, 3249792, 6604416, 20321280, 104993280, 1889879040, 37797580800, 907141939200, 1828682956800, 54860488704000, 1755535638528000, 12508191424512000, 37837279059148800, 1437816604247654400, 60388297378401484800

Offset: 1

Arkadiusz Wesolowski, Aug 18 2012

nonn

			6 is the second colossally abundant number. Divisors of 6 are 1, 2, 3, 6, so a(2) = 1 + 2 + 3 + 6 = 12.

Amiram Eldar, Table of n, a(n) for n = 1..382 (terms 1..128 from Arkadiusz Wesolowski)
Eric Weisstein's World of Mathematics, Riemann Hypothesis.

Cf. A000203, A004490, A058209, A080130, A207709.

lst1 = {2}; lst2 = {}; maxN = 23; p = 1; pFactor[f_List] := Module[{p = f[[1]], k = f[[2]]}, N[Log[(p^(k + 2) - 1)/(p^(k + 1) - 1)]/Log[p]] - 1]; f = {{2, 1}, {3, 0}}; primes = 1; x = Table[pFactor[f[[i]]], {i, primes + 1}]; For[n = 2, n <= maxN, n++, i = Position[x, Max[x]][[1, 1]]; AppendTo[lst1, f[[i, 1]]]; f[[i, 2]]++; If[i > primes, primes++; AppendTo[f, {Prime[i + 1], 0}]; AppendTo[x, pFactor[f[[-1]]]]]; x[[i]] = pFactor[f[[i]]]]; Do[p = p*lst1[[n]]; AppendTo[lst2, DivisorSigma[1, p]], {n, maxN}]; lst2 (* Most of the code is from T. D. Noe *)

a(n) = A000203(A004490(n)).

cp's OEIS Frontend

A218342 Decimal expansion of e^-gamma * Product_(1 - 1/(p^3 - p^2 - p + 1)) where the product is over all primes p.

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A335636 Expansion of e.g.f. Product_{k>0} 1/(1 - tan(x)^k / k).

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A335638 Expansion of e.g.f. Product_{k>0} (1 + tan(x)^k / k).

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A368246 Number of permutations of [n] whose cycle minima sum to n.

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A242909 Decimal expansion of exp(-gamma/2).

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A244274 Decimal expansion of e*gamma, the product of Euler number and Euler-Mascheroni constant.

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A246499 Decimal expansion of zeta(2)/exp(gamma), gamma being the Euler-Mascheroni constant.

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