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 1-4 of 4 results.

A072155 Denominator of Sum_{k=1..n} phi(k)/k.

Original entry on oeis.org

1, 2, 6, 3, 15, 5, 35, 70, 210, 210, 2310, 770, 10010, 10010, 30030, 15015, 255255, 255255, 4849845, 4849845, 4849845, 4849845, 111546435, 37182145, 37182145, 37182145, 111546435, 111546435, 3234846615, 3234846615, 100280245065, 200560490130, 200560490130
Offset: 1

Views

Author

N. J. A. Sloane, Jun 28 2002

Keywords

Examples

			1, 3/2, 13/6, 8/3, 52/15, 19/5, 163/35, 361/70, 1223/210, ...

Links

Crossrefs

Cf. A071708 (numerators), A250031, A250032, A250034.

Programs

  • GAP
    List([1..35], n-> DenominatorRat( Sum([1..n], k-> Phi(k)/k) ) ); # G. C. Greubel, Aug 25 2019
  • Magma
    [Denominator( &+[EulerPhi(k)/k: k in [1..n]] ): n in [1..35]]; // G. C. Greubel, Aug 25 2019
  • Maple
    with(numtheory); seq(denom( add(phi(k)/k, k=1..n)), n =1..35); # G. C. Greubel, Aug 25 2019
  • Mathematica
    Table[Sum[EulerPhi[k]/k,{k,n}],{n,40}]//Denominator (* Harvey P. Dale, Jun 08 2017 *)
  • PARI
    a(n) = denominator(sum(k=1, n, eulerphi(k)/k)); \\ Michel Marcus, Jan 26 2015
  • Sage
    [denominator( sum(euler_phi(k)/k for k in (1..n)) ) for n in (1..35)] # G. C. Greubel, Aug 25 2019

Formula

Also denominator of Sum_{i=1..n} (mu(i)/i)*floor(n/i). - Ridouane Oudra, Nov 26 2019

A333695 Numerators of coefficients in expansion of Sum_{k>=1} phi(k) * log(1/(1 - x^k)).

Original entry on oeis.org

1, 3, 7, 11, 21, 7, 43, 43, 61, 63, 111, 77, 157, 129, 49, 171, 273, 61, 343, 231, 43, 333, 507, 301, 521, 471, 547, 473, 813, 147, 931, 683, 259, 819, 129, 671, 1333, 1029, 1099, 903, 1641, 43, 1807, 111, 427, 1521, 2163, 399, 2101, 1563, 637, 1727, 2757, 547, 2331
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 02 2020

Keywords

Examples

			1, 3/2, 7/3, 11/4, 21/5, 7/2, 43/7, 43/8, 61/9, 63/10, 111/11, 77/12, 157/13, 129/14, 49/5, ...

Crossrefs

Cf. A000010, A000203, A001157, A018804, A057660, A071708, A072155, A074947, A074949, A333696 (denominators), A346602.

Programs

  • Mathematica
    nmax = 55; CoefficientList[Series[Sum[EulerPhi[k] Log[1/(1 - x^k)], {k, 1, nmax}], {x, 0, nmax}], x] // Numerator // Rest
Table[Sum[EulerPhi[n/d]/d, {d, Divisors[n]}], {n, 55}] // Numerator
Table[Sum[1/GCD[n, k], {k, n}], {n, 55}] // Numerator
Table[DivisorSigma[2, n^2]/(n DivisorSigma[1, n^2]), {n, 55}] // Numerator
  • PARI
    a(n) = numerator(sumdiv(n, d, eulerphi(n/d) / d)); \\ Michel Marcus, Apr 03 2020

Formula

a(n) = numerator of Sum_{d|n} phi(n/d) / d.
a(n) = numerator of Sum_{k=1..n} 1 / gcd(n,k).
a(n) = numerator of sigma_2(n^2) / (n * sigma_1(n^2)).
a(p) = p^2 - p + 1 where p is prime.
From Amiram Eldar, Nov 21 2022: (Start)
a(n) = numerator(A057660(n)/n).
Sum_{k=1..n} a(k)/A333696(k) ~ c * n^2, where c = zeta(3)/(2*zeta(2)) = 0.365381... (A346602). (End)

A333696 Denominators of coefficients in expansion of Sum_{k>=1} phi(k) * log(1/(1 - x^k)).

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 7, 8, 9, 10, 11, 12, 13, 14, 5, 16, 17, 6, 19, 20, 3, 22, 23, 24, 25, 26, 27, 28, 29, 10, 31, 32, 11, 34, 5, 36, 37, 38, 39, 40, 41, 2, 43, 4, 15, 46, 47, 16, 49, 50, 17, 52, 53, 18, 55, 56, 57, 58, 59, 20, 61, 62, 63, 64, 65, 22, 67, 68, 23, 10
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 02 2020

Keywords

Examples

			1, 3/2, 7/3, 11/4, 21/5, 7/2, 43/7, 43/8, 61/9, 63/10, 111/11, 77/12, 157/13, 129/14, 49/5, ...

Crossrefs

Cf. A000010, A000203, A001157, A018804, A057660, A070999, A071000 (fixed points), A071708, A072155, A074947, A074949, A333695 (numerators).

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[Sum[EulerPhi[k] Log[1/(1 - x^k)], {k, 1, nmax}], {x, 0, nmax}], x] // Denominator // Rest
Table[Sum[EulerPhi[n/d]/d, {d, Divisors[n]}], {n, 70}] // Denominator
Table[Sum[1/GCD[n, k], {k, n}], {n, 70}] // Denominator
Table[DivisorSigma[2, n^2]/(n DivisorSigma[1, n^2]), {n, 70}] // Denominator
  • PARI
    a(n) = denominator(sumdiv(n, d, eulerphi(n/d) / d)); \\ Michel Marcus, Apr 03 2020

Formula

a(n) = denominator of Sum_{d|n} phi(n/d) / d.
a(n) = denominator of Sum_{k=1..n} 1 / gcd(n,k).
a(n) = denominator of sigma_2(n^2) / (n * sigma_1(n^2)).

A385561 Numbers m such that (1/m) * Sum_{k=1..m} phi(k)/k is closer to 6/Pi^2 than it is for any number smaller than m, where phi is the Euler totient function (A000010).

Original entry on oeis.org

1, 2, 3, 4, 6, 10, 12, 16, 22, 28, 36, 66, 96, 100, 126, 156, 190, 330, 430, 540, 820, 876, 1086, 1422, 10596, 10836, 18096, 35796, 55786, 69336, 111100, 168666, 284650, 905950, 1482300, 1745590, 2405560, 2661310, 4023306, 5869956, 17454580, 25670646, 51305346, 79969618, 211025650, 622626790
Offset: 1

Views

Author

Amiram Eldar, Jul 03 2025

Keywords

Comments

6/Pi^2 is the asymptotic mean of phi(k)/k, i.e., lim_{m->oo} (1/m) * Sum_{k=1..m} phi(k)/k = 6/Pi^2 (Walfisz, 1963; Sándor et al., 2005).

References

  • József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 27.
  • Arnold Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Berlin, 1963.

Crossrefs

Cf. A000010, A059956, A071708, A072155, A330899, A385562.

Programs

  • Mathematica
    seq[lim_] := Module[{s = {}, sum = 0, dm = 1, d}, Do[sum += EulerPhi[k]/k; If[(d = Abs[sum/k - 6/Pi^2]) < dm, dm = d; AppendTo[s, k]], {k, 1, lim}]; s]; seq[10^5]
  • PARI
    list(lim) = {my(sm = 0, dm = 1, d); for(k = 1, lim, sm += eulerphi(k)/k; d = abs(sm/k - 6/Pi^2); if(d < dm, dm = d; print1(k, ", ")));}
Showing 1-4 of 4 results.

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

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