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.

A065393 Sigma(phi(m)) - phi(sigma(m)) is increasing at these values of m.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 29, 31, 37, 41, 53, 61, 73, 95, 97, 109, 127, 143, 157, 181, 209, 241, 287, 313, 323, 337, 377, 403, 407, 421, 473, 527, 533, 541, 589, 601, 661, 713, 731, 757, 779, 899, 1009, 1073, 1147, 1159, 1199, 1271, 1321, 1333, 1349, 1517
Offset: 1

Views

Author

Labos Elemer, Nov 05 2001

Keywords

Comments

First composite number is the 15th term, 95. [Corrected by Jacob Vecht, Jul 28 2020]

Links

Crossrefs

Cf. A000010, A000203, A065394, A065395.

Programs

  • Mathematica
    a = 0; s = 0; Do[s = DivisorSigma[1, EulerPhi[n]] - EulerPhi[DivisorSigma[1, n]]; If[s>a, a = s; Print[n]], {n, 1, 10000}]
DeleteDuplicates[Table[{m,DivisorSigma[1,EulerPhi[m]]-EulerPhi[DivisorSigma[1,m]]},{m,1600}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Aug 02 2023 *)
  • PARI
    { n=r=0; for (m=1, 10^9, x=sigma(eulerphi(m)) - eulerphi(sigma(m)); if (x > r, r=x; write("b065393.txt", n++, " ", m); if (n==500, return)) ) } \\ Harry J. Smith, Oct 18 2009

A065392 a(n) = A062401(A065391(n)): phi(sigma(m)) peak values for numbers m (listed in A065391) at which those peaks are first reached.

Original entry on oeis.org

1, 2, 6, 8, 12, 30, 36, 72, 126, 180, 360, 432, 660, 930, 1512, 2160, 3300, 3780, 5184, 6552, 11160, 13860, 19800, 23232, 32760, 45360, 47520, 50400, 58080, 61776, 102300, 110160, 137592, 155520, 163296, 196560, 212960, 252000, 272160, 284580
Offset: 1

Views

Author

Labos Elemer, Nov 14 2001

Keywords

Links

Crossrefs

Cf. A000010, A000203, A062401, A065389, A065390, A065391, A065393, A065394.

Programs

  • Mathematica
    With[{s = Array[EulerPhi@ DivisorSigma[1, #] &, 10^5]}, Union@ FoldList[Max, s] ] (* Michael De Vlieger, Dec 06 2018 *)
  • PARI
    { n=r=0; for (m=1, 10^9, x=eulerphi(sigma(m)); if (x > r, r=x; write("b065392.txt", n++, " ", x); if (n==100, return)) ) } \\ Harry J. Smith, Oct 18 2009

A065390 Peak values reached by A062402 at the sites listed in A065389.

Original entry on oeis.org

1, 3, 7, 12, 18, 28, 31, 39, 42, 56, 72, 91, 96, 98, 168, 195, 252, 280, 312, 360, 372, 392, 546, 576, 744, 840, 864, 992, 1092, 1170, 1344, 1512, 1680, 1860, 1872, 2016, 2240, 2418, 2880, 3224, 3600, 3844, 4320, 4368, 4914, 5082, 5952, 6045, 6552, 7440
Offset: 1

Views

Author

Labos Elemer, Nov 05 2001

Keywords

Links

Crossrefs

Cf. A062402, A000203, A000010, A065389, A065391, A065392, A065393, A065394.

Programs

  • Mathematica
    a=0; s=0; Do[s=DivisorSigma[1, EulerPhi[n]]; If[s>a, a=s; Print[s]], {n, 1, 10000}]
(* Second program: *)
Union@ FoldList[Max, Array[DivisorSigma[1, EulerPhi[#]] &, 2200]] (* Michael De Vlieger, Jun 19 2018 *)
  • PARI
    { n=r=0; for (m=1, 10^9, x=sigma(eulerphi(m)); if (x > r, r=x; write("b065390.txt", n++, " ", x); if (n==500, return)) ) } \\ Harry J. Smith, Oct 18 2009

Formula

a(n) = A062402(A065389(n)). - Amiram Eldar, Mar 22 2025

A092590 a(n) = A065395(A000040(n)); values of commutator of sigma and phi function at prime number arguments.

Original entry on oeis.org

-1, 1, 5, 8, 14, 22, 25, 31, 28, 48, 56, 73, 78, 76, 56, 80, 74, 138, 112, 120, 159, 136, 102, 156, 210, 185, 168, 126, 240, 212, 248, 212, 226, 240, 226, 300, 314, 283, 204, 252, 222, 474, 296, 412, 339, 388, 472, 360, 270, 472, 378, 368, 634, 396, 427, 316, 404, 592, 534, 628, 436, 434, 582, 480, 684, 456, 700, 836
Offset: 1

Views

Author

Labos Elemer, Mar 03 2004

Keywords

Comments

The sequence differs from A065394 since it is not monotonic.

Examples

			a(1) = sigma(phi(2))- phi(sigma(2)) = sigma(1)-phi(3) = 1-2 = -1.

Links

Crossrefs

Cf. A000010, A000040, A000203, A033632, A065393, A065394, A065395, A008331, A008332, A092584, A092585, A092586, A092587, A092588, A092589.

Programs

  • Magma
    [DivisorSigma(1,EulerPhi(p))-EulerPhi(DivisorSigma(1,p)): p in PrimesUpTo(400)]; // Bruno Berselli, Oct 20 2015
  • Mathematica
    Table[DivisorSigma[1, p-1] - EulerPhi[p+1], {p, Prime[Range[100]]}] (* Amiram Eldar, Jun 09 2024 *)

Formula

a(n) = sigma(prime(n)-1) - phi(prime(n)+1) = A008332(n) - A008331(n). - Amiram Eldar, Jun 09 2024
Showing 1-4 of 4 results.

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