A085609 -id:A085609 - cp's OEIS frontend

A028415 Numerator of Sum_{k=1..n} 1/phi(k).

1, 2, 5, 3, 13, 15, 47, 25, 13, 55, 281, 74, 301, 311, 637, 163, 1319, 453, 4117, 4207, 4267, 4339, 48089, 49079, 9895, 10027, 10115, 10247, 72125, 73511, 369403, 93217, 9391, 75821, 76283, 77207, 77515, 78131, 78593, 39643, 49727, 100609, 100939, 25408, 204419

Offset: 1

N. J. A. Sloane, Jun 28 2002

			1, 2, 5/2, 3, 13/4, 15/4, 47/12, 25/6, 13/3, 55/12, 281/60, 74/15, ...

József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Section I.27, page 29.

Robert Israel, Table of n, a(n) for n = 1..10000
R. Sitaramachandrarao, On an error term of Landau - II, The Rocky Mountain Journal of Mathematics, Vol. 15, No. 2 (1985), pp. 579-588.
Eric Weisstein's World of Mathematics, Totient Summatory Function.

Cf. A000010, A048049 (denominators).

Cf. A001620, A085609, A082695.

map(numer, ListTools:-PartialSums(map(1/numtheory:-phi, [$1..10000]))); # Robert Israel, Apr 16 2019

Numerator[Table[Sum[1/EulerPhi[k],{k,n}],{n,50}]] (* Harvey P. Dale, Aug 24 2012 *)

a(n) = numerator(sum(k=1, n, 1/eulerphi(k))); \\ Michel Marcus, Sep 18 2022

a(n)/A048049(n) = c * (log(n) + gamma - s) + O(log(n)^(2/3)/n), where c = zeta(2)*zeta(3)/zeta(6) (A082695), gamma is Euler's constant (A001620), and s = Sum_{p prime} log(p)/(p^2-p+1) (A085609) (Sitaramachandrarao, 1985). - Amiram Eldar, Sep 18 2022

A306072 Decimal expansion of 2 * Sum_{p prime}(p^2-p-1)log(p)/(p^4+2p^3+1).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jun 19 2018

The constant B that appears in the asymptotic formula for the sum of the bi-unitary divisor function (A306069).

			0.405237031444223925085965099112185234104414172404198492623463629775379...

Cf. A085609, A306069, A335705, A335707.

cc = CoefficientList[Series[(p^2 - p - 1)/(p^4 + 2*p^3 + 1) /. p -> 1/x, {x, 0, 30}], x]; f = FindSequenceFunction[cc]; digits = 20; B = 2 NSum[f[n + 1 // Round]*(-PrimeZetaP'[n]), {n, 2, Infinity}, Method -> "AlternatingSigns", NSumTerms -> 10 digits, WorkingPrecision -> 5 digits]; RealDigits[B, 10, digits][[1]] (* Jean-François Alcover, Jun 19 2018 *)
ratfun = 2*(p^2 - p - 1)/(p^4 + 2*p^3 + 1); zetas = 0; ratab = Table[konfun = Simplify[ratfun + c/(p^power - 1)] // Together; coefs = CoefficientList[Numerator[konfun], p]; sol = Solve[Last[coefs] == 0, c][[1]]; zetas = zetas + c*Zeta'[power]/Zeta[power] /. sol; ratfun = konfun /. sol, {power, 2, 20}]; Do[Print[N[Sum[Log[p]*ratfun /. p -> Prime[k], {k, 1, m}] + zetas, 100]], {m, 2000, 20000, 2000}] (* Vaclav Kotesovec, Jun 17 2020 *)

a(1)-a(20) from Jean-François Alcover, Jun 19 2018

More digits from Vaclav Kotesovec, Jun 17 2020

A335707 Decimal expansion of Sum_{primes p} log(p) / (p^2 + p - 1).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Jun 18 2020

			0.41875757879412548053442125602870463613655516544928702940522...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 100, p. 169.

Cf. A062355, A085609.

ratfun = 1 / (p^2 + p - 1); zetas = 0; ratab = Table[konfun = Simplify[ratfun + c/(p^power - 1)] // Together; coefs = CoefficientList[Numerator[konfun], p]; sol = Solve[Last[coefs] == 0, c][[1]]; zetas = zetas + c*Zeta'[power]/Zeta[power] /. sol; ratfun = konfun /. sol, {power, 2, 20}]; Do[Print[N[Sum[Log[p]*ratfun /. p -> Prime[k], {k, 1, m}] + zetas, 100]], {m, 2000, 20000, 2000}]

A211177 Numerator of Sum_{k=1..n}(-1)^k/phi(k), where phi = A000010.

Original entry on oeis.org

-1, 0, -1, 0, -1, 1, 1, 1, 1, 5, 19, 17, 29, 13, 21, 13, 47, 181, 503, 593, 533, 121, 1259, 1457, 6889, 7549, 7109, 7769, 52403, 59333, 11497, 6095, 29089, 61643, 59333, 63953, 62413, 7277, 21061, 2777, 10877, 11647, 3809, 3963, 1438, 271, 3064, 51439, 7217, 7493

Offset: 1

Benoit Cloitre, Feb 01 2013

			Fractions begin with -1, 0, -1/2, 0, -1/4, 1/4, 1/12, 1/3, 1/6, 5/12, 19/60, 17/30, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
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.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A000010, A028415, A211178 (denominators).

Cf. A001620, A082695, A085609.

Numerator @ Accumulate[Table[(-1)^k/EulerPhi[k], {k, 1, 50}]] (* Amiram Eldar, Nov 20 2020 *)

a(n)=numerator(sum(k=1,n,(-1)^k/eulerphi(k)))

a(n)/A211178(n) = c*log(n) + O(1) with a suitable constant c (see ref).
The constant above is c = zeta(2)*zeta(3)/(3*zeta(6)) = (1/3) * A082695. - Amiram Eldar, Nov 20 2020
More accurately, a(n)/A211178(n) ~ (A/3) * (log(n) + gamma - B - 8*log(2)/3) + O(log(n)^(5/3)/n), where A = zeta(2)*zeta(3)/zeta(6) (A082695), gamma is Euler's constant (A001620), and B = Sum_{p prime} log(p)/(p^2-p+1) (A085609) (Bordellès and Cloitre, 2013; Tóth, 2017). - Amiram Eldar, Oct 14 2022

More terms from Amiram Eldar, Nov 20 2020

A345364 Decimal expansion of Sum_{p primes} p * (log(p))^2 / (p-1)^3.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Jun 16 2021

			2.0914802823489018573384036648086053401546322612324184299409135322256726453113...

Eric Weisstein's World of Mathematics, Prime Factor, formula (15)-(16), (constant V).

Cf. A085609, A306072, A335705, A335706, A335707, A345308.

ratfun = p/(p - 1)^3; zetas = 0; ratab = Table[konfun = Together[Simplify[ratfun - c*(p^power/(p^power - 1)^2)]]; coefs = CoefficientList[Numerator[konfun], p]; sol = Solve[Last[coefs] == 0, c][[1]]; zetas = zetas + c*(-Zeta'[power]^2 / Zeta[power]^2 + Zeta''[power] / Zeta[power]) /. sol; ratfun = konfun /. sol, {power, 2, 30}]; Do[Print[N[Sum[Log[p]^2*ratfun /. p -> Prime[k], {k, 1, m}] + zetas, 110]], {m, 100, 1000, 100}]

A074467 Least k such that Sum_{i=1..k} 1/phi(i) >= n.

Original entry on oeis.org

1, 2, 4, 8, 13, 22, 38, 63, 105, 177, 296, 495, 828, 1386, 2318, 3879, 6489, 10854, 18158, 30375, 50811, 84998, 142187, 237853, 397885, 665589, 1113411, 1862534, 3115683, 5211973, 8718687, 14584783, 24397699, 40812930, 68272636, 114207749, 191048868, 319590137

Offset: 1

Labos Elemer, Aug 29 2002

nonn

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 177, p. 55, Ellipses, Paris 2008.
E. Landau, Uber die Zahlentheoretische Function ϕ(n) und ihre Beziehung zum Goldbachschen satz, Nachrichten der Koniglichten Gesel lschaft der Wissenschaften zu Göttingen mathematisch Physikalische klasse, Jahrgang (1900), pp. 177-186.

H. L. Montgomery, Primes in arithmetic progressions, Michigan Math. J. 17:1 (1970), pp. 33-39. [alternate link]

Cf. A002387, A004080, A046024, A074631, A074633.

{s=0, s1=0}; Do[s=s+(1/EulerPhi[n]); If[Greater[Floor[s], s1], s1=Floor[s]; Print[{n, Floor[s]}]], {n, 1, 1000000}]

a(n)=my(s,k);while(sCharles R Greathouse IV, Jan 29 2013

a(n) ~ k exp(cn) for c = zeta(6)/zeta(2)/zeta(3) = A068468 and k = exp(-gamma + A085609) = 1.0316567993311528...; see Montgomery or Koninck. - Charles R Greathouse IV, Jan 29 2013

More terms from Ryan Propper, Jul 09 2005

a(32)-a(38) from Donovan Johnson, Aug 21 2011

A098468 Decimal expansion of constant A*B in the asymptotic expression of the summatory function Sum_{n=1..N} (1/phi(n)) as A(log(N)+B) + O(log(N)/N).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Sep 09 2004

B equals EulerGamma - A085609.

			B = -0.0605742294.../A, where A is A082695.

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.7 Euler totient constants, p. 116.

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 52 (Z1*(gamma-Z2)).
Eric Weisstein's World of Mathematics, Totient Summatory Function

Cf. A082695, A085609.

(* Using S. Finch's notation *)
digits = 102;
A = Zeta[2]*Zeta[3]/Zeta[6];
S = Sum[Switch[Mod[k, 6], 0, 1, 1, 0, 2, -1, 3, -1, 4, 0, 5, 1]*PrimeZetaP'[k], {k, 2, 400}] // N[#, digits+40]&;
B = EulerGamma - S;
AB = A*B;
Join[{0}, RealDigits[AB, 10, digits][[1]]] (* Jean-François Alcover, Apr 28 2018 *)

Sum_{n=1..N} 1/phi(n) = A*(log(N)+B) + O(log(N)/N). - Jean-François Alcover, Apr 28 2018

More digits with the aid of A085609 and A082695 from R. J. Mathar, Jul 28 2010

More digits with the aid of A085609 and A082695 from Vaclav Kotesovec, Feb 17 2015

A373863 Decimal expansion of Sum_{k>=1} log(k)/(k^2-k+1).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jun 19 2024

			1.0981067917544222069517...

Cf. A085609, A131688.

g := (1-e+e^2)^(-1) ;
x :=0.0 ;
for i from 0 to 350 do
    coeftayl(g,e=0,i) ;
    x-%*Zeta(1,2+i) ;
    x := evalf(%) ;
    print(%) ;
end do:

default(realprecision, 200); sumpos(k=1, log(k+1)/(k^2+k+1)) \\ Vaclav Kotesovec, Jun 28 2024

Equals Sum_{k>=1} log(k+1)/(k^2+k+1).

cp's OEIS Frontend

A028415 Numerator of Sum_{k=1..n} 1/phi(k).

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A306072 Decimal expansion of 2 * Sum_{p prime}(p^2-p-1)*log(p)/(p^4+2*p^3+1).

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A335707 Decimal expansion of Sum_{primes p} log(p) / (p^2 + p - 1).

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A211177 Numerator of Sum_{k=1..n}(-1)^k/phi(k), where phi = A000010.

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A345364 Decimal expansion of Sum_{p primes} p * (log(p))^2 / (p-1)^3.

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A074467 Least k such that Sum_{i=1..k} 1/phi(i) >= n.

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A098468 Decimal expansion of constant A*B in the asymptotic expression of the summatory function Sum_{n=1..N} (1/phi(n)) as A(log(N)+B) + O(log(N)/N).

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A373863 Decimal expansion of Sum_{k>=1} log(k)/(k^2-k+1).

A306072 Decimal expansion of 2 * Sum_{p prime}(p^2-p-1)log(p)/(p^4+2p^3+1).