A074633 -id:A074633 - cp's OEIS frontend

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

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

A074468 Least number m such that the Sigma-Harmonic sequence Sum_{k=1..m} 1/sigma(k) >= n.

Original entry on oeis.org

1, 7, 29, 129, 571, 2525, 11167, 49372, 218295, 965177, 4267457, 18868240, 83424514, 368855252, 1630865929, 7210751807, 31881800153

Offset: 1

Labos Elemer, Aug 29 2002

Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 129, p. 44, Ellipses, Paris, 2008.

Cf. A000203 (sigma), A002387, A004080, A046024, A074631, A074633, A074467, A212717, A308039.

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

Limit_{n->oo} a(n+1)/a(n) = exp(1/c) = 4.42142525588146107878... where c = A308039. - Amiram Eldar, May 05 2024

2 more terms from Lekraj Beedassy, Jul 14 2008
a(11)-a(15) from Donovan Johnson, Aug 22 2011
a(16)-a(17) from Amiram Eldar, May 05 2024

A074469 Least m such that Sigma-Composite-Harmonic series Sum_{k=1..m} 1/A000203(A002808(k)) >= n.

Original entry on oeis.org

32, 301, 2123, 13172, 76105, 420007, 2245009, 11719362, 60071831, 303487314, 1515211979

Offset: 1

Labos Elemer, Sep 05 2002

Cf. A004080, A002387, A002808, A000203, A073255, A046024, A074631, A074633, A074468, A074470.

c[x_] := FixedPoint[x+PrimePi[ # ]+1&, x] {s=0, s1=0}; Do[s=s+(1/DivisorSigma[1, c[n]]); If[Greater[Floor[s], s1], s1=Floor[s]; Print[{n, Floor[s]}]], {n, 1, 1000000}]

a(n)=my(m,s=0.);for(c=4,(2*n+2)^(n+2),if(isprime(c),next,m++);s+=1/sigma(c);if(s>=n,return(m))) \\ Charles R Greathouse IV, Feb 19 2013

a(6)-a(11) from Donovan Johnson, Aug 22 2011

A074470 Least m such that Phi-Composite-Harmonic series Sum_{k=1..m} 1/A000010(A002808(k)) >= n.

Original entry on oeis.org

2, 7, 16, 31, 60, 113, 205, 371, 663, 1176, 2069, 3631, 6341, 11039, 19159, 33164, 57287, 98763, 169967, 292061, 501165, 858892, 1470334, 2514423, 4295912, 7333264, 12508213, 21319360, 36312685, 61811287, 105152840, 178787270, 303829041, 516074615, 876190239

Offset: 1

Labos Elemer, Sep 05 2002

Cf. A004080, A002387, A002808, A000203, A073255, A046024, A074631, A074633, A074468, A074469.

c[x_] := FixedPoint[x+PrimePi[ # ]+1&, x] {s=0, s1=0}; Do[s=s+(1/EulerPhi[c[n]]); If[Greater[Floor[s], s1], s1=Floor[s]; Print[{n, Floor[s]}]], {n, 1, 1000000}]

More terms from Lambert Klasen (lambert.klasen(AT)gmx.net), Jul 23 2005

a(30)-a(35) from Donovan Johnson, Aug 21 2011

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

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A074468 Least number m such that the Sigma-Harmonic sequence Sum_{k=1..m} 1/sigma(k) >= n.

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A074469 Least m such that Sigma-Composite-Harmonic series Sum_{k=1..m} 1/A000203(A002808(k)) >= n.

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A074470 Least m such that Phi-Composite-Harmonic series Sum_{k=1..m} 1/A000010(A002808(k)) >= n.

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