cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A074633 a(n) is the smallest index m such that Sum_{k=2..m} 1/PrimePi(k) >= n, where PrimePi()=A000720().

Original entry on oeis.org

2, 4, 8, 12, 18, 27, 37, 51, 68, 89, 116, 147, 186, 232, 287, 352, 428, 518, 623, 745, 887, 1050, 1240, 1456, 1704, 1987, 2309, 2674, 3090, 3557, 4087, 4684, 5353, 6105, 6949, 7892, 8944, 10121, 11431, 12885, 14502, 16298, 18286, 20485, 22917, 25607
Offset: 1

Views

Author

Labos Elemer, Aug 28 2002

Keywords

Examples

			a(85) = 927685 because 927686 is the smallest m such that Sum_{k=2..m} 1/PrimePi(k) >= 85.

Crossrefs

Cf. A004080, A046024, A074631, A000720.

Programs

  • Mathematica
    {s=0, s1=0}; Do[s=s+(1/PrimePi[n]); If[Greater[Floor[s], s1], s1=Floor[s]; Print[{n, Floor[s]}]], {n, 2, 1000000}]

Extensions

Edited by Jon E. Schoenfield, Apr 04 2023
Name corrected by Sean A. Irvine, Jan 22 2025

A076751 a(n) is the smallest composite k such that Sum_{composites j = 4, ..., k} 1/j exceeds n.

Original entry on oeis.org

16, 63, 216, 715, 2279, 7102, 21722, 65558, 195759, 579465, 1703072, 4975222, 14459492, 41837580, 120585504, 346372172, 991915208, 2832896772, 8071045528, 22944211170
Offset: 1

Views

Author

Jack Brennen, Nov 12 2002

Keywords

Comments

These partial sums, like the harmonic sequence (A004080), can never be integers.

Examples

			a(1) = 1 because 1/4 + 1/6 + 1/8 + 1/9 + 1/10 + 1/12 + 1/14 + 1/15 = 0.97420... < 1 but 1/4 + 1/6 + 1/8 + 1/9 + 1/10 + 1/12 + 1/14 + 1/15 + 1/16 = 1.03670... > 1.

Crossrefs

Cf. A002808, A004080, A016088, A074631.

Programs

  • Mathematica
    NextComposite[n_] := Block[{k = n + 1}, While[ PrimeQ[k], k++ ]; k]; k = 4; s = 0; Do[ While[s = s + 1/k; s < n, k = NextComposite[k]]; Print[k]; k = NextComposite[k], {n, 1, 17}]
  • PARI
    lista(cmax) = {my(n = 1, s = 0); forcomposite(c = 1, cmax, s += 1/c; if(s > n, print1(c, ", "); n++));} \\ Amiram Eldar, Jul 17 2024

Formula

Limit_{n->oo} a(n+1)/a(n) = e.
a(n) = A002808(A074631(n)). - Amiram Eldar, Jul 17 2024

Extensions

Edited and extended by Robert G. Wilson v, Nov 14 2002
Name edited and a(18) added by Jon E. Schoenfield, Feb 01 2020
a(19)-a(20) from Amiram Eldar, Jul 17 2024

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

Views

Author

Labos Elemer, Aug 29 2002

Keywords

References

  • 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.

Links

Crossrefs

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

Programs

  • Mathematica
    {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}]
  • PARI
    a(n)=my(s,k);while(sCharles R Greathouse IV, Jan 29 2013

Formula

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

Extensions

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

Views

Author

Labos Elemer, Aug 29 2002

Keywords

References

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

Crossrefs

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

Programs

  • Mathematica
    {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}]

Formula

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

Extensions

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

A376934 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, Sum_{k = 1..n} 1/(k*a(k)) < Pi.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 6, 80, 21896, 604163887, 677040508659246685, 447405708743254015046365510044832005, 309471557529368331206803181535934923519436869019793750609292014082198479
Offset: 1

Views

Author

Scott R. Shannon, Oct 11 2024

Keywords

Examples

			a(17) = 604163887 as Sum_{k = 1..17} 1/(k*a(k)) = 1/(1*1) + 1/(2*1) + ... + 1/(16*21896) + 1/(17*604163887) = 3056398076673607759/972881723918332800, which is ~8.2*10^-20 less than Pi.

Links

Crossrefs

Cf. A000796, A270744, A375781, A375529, A074631, A270752.

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

Views

Author

Labos Elemer, Sep 05 2002

Keywords

Crossrefs

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

Programs

  • Mathematica
    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}]
  • PARI
    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

Extensions

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

Views

Author

Labos Elemer, Sep 05 2002

Keywords

Crossrefs

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

Programs

  • Mathematica
    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}]

Extensions

More terms from Lambert Klasen (lambert.klasen(AT)gmx.net), Jul 23 2005
a(30)-a(35) from Donovan Johnson, Aug 21 2011

A376941 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, Sum_{k = 1..n} 1/(k*a(k)) < Sum_{k = 1..oo} 1/k^2 = Pi^2/6.

Original entry on oeis.org

1, 1, 3, 8, 78, 18826, 848809436, 3078251288697343844, 37949774653961921717972183675013581047, 4221065664206126654046840821317530741151656832301121739091602085731664210467
Offset: 1

Views

Author

Scott R. Shannon, Oct 12 2024

Keywords

Examples

			a(7) = 848809436 as Sum_{k = 1..7} 1/(k*a(k)) = 1/(1*1) + 1/(2*1) + ... + 1/(6*18826) + 1/(7*848809436) = 430556991329920237/261747263922187680, which is ~4.1*10^-20 less than Pi^2/6.

Links

Crossrefs

Cf. A013661, A376934, A270744, A375781, A375529, A074631, A270752.
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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