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.

A083250 Numbers n for which abs(A045763(n) - A073757(n)) = 3, i.e., signed difference of size of related and unrelated sets to n equals either 3 or -3.

Original entry on oeis.org

3, 825, 1755, 4641, 5313, 56865, 58395, 140049, 159152763, 361701435
Offset: 1

Views

Author

Labos Elemer, May 07 2003

Keywords

Comments

No more terms below 10^9. - Amiram Eldar, Jan 27 2019

Examples

			For n=140049: d=40 divisors, r=70026 coprimes and u=70023 unrelated numbers to n; abs(r-u) = 3.

Crossrefs

Cf. A000005, A000010, A045763, A073757, A083243-A083249.

Programs

  • Mathematica
    Do[r=EulerPhi[n]; d=DivisorSigma[0, n]; u=n-r-d+1; df=2*u-n; If[Equal[Abs[df], 3], Print[n(*, {d, r, u}*)]], {n, 1, 100}]
  • PARI
    isok(n) = abs(n-2*eulerphi(n)-2*numdiv(n)+2) == 3; \\ Michel Marcus, Jul 30 2017

Extensions

a(9)-a(10) from Amiram Eldar, Jan 27 2019

A083251 Numbers n such that abs(A045763(n) - A073757(n)) = 2, i.e., signed difference of size of related and unrelated sets to n equals either 2 or -2.

Original entry on oeis.org

2, 48, 72, 80, 112, 176, 208, 272, 304, 368, 464, 496, 592, 656, 688, 752, 848, 944, 976, 1072, 1136, 1168, 1264, 1328, 1424, 1552, 1616, 1648, 1712, 1744, 1808, 2032, 2096, 2192, 2224, 2384, 2416, 2512, 2608, 2672, 2768, 2864, 2896
Offset: 1

Views

Author

Labos Elemer, May 07 2003

Keywords

Examples

			For n=2896: d=10 divisors, r=1440 coprimes, u=1447 unrelated or n - u = r + d - 1 = 1449 related numbers to n; thus abs(1449 - 1447) = 2.

Links

Crossrefs

Cf. A000005, A000010, A045763, A073757, A083243-A083249, A083250.

Programs

  • Mathematica
    Do[r=EulerPhi[n]; d=DivisorSigma[0, n]; u=n-r-d+1; df=2*u-n; If[Equal[Abs[df], 2], Print[n(*, {d, r, u}*)]], {n, 1, 3000}]

Formula

a(n) = 8 * (A076274(n-1) + 1) for n > 3, as proved by Lawrence Sze. - Ralf Stephan, Nov 16 2004

A083252 Numbers k for which abs(A045763(k) - A073757(k)) = 5, i.e., signed difference of size of related and unrelated sets to k equals either 5 or -5.

Original entry on oeis.org

5, 105, 315, 182835, 960075, 7838265, 4291166265
Offset: 1

Views

Author

Labos Elemer, May 07 2003

Keywords

Comments

a(7), if it exists, is > 10^9. - Vaclav Kotesovec, Sep 06 2019

Examples

			For k = 960075: d = 36 divisors, r = 480000 coprimes, u = 480040 unrelated; k - u = r + d - 1 = 480035 related numbers to k; thus abs(480040 - 480035) = 5.

Crossrefs

Cf. A000005, A000010, A045763, A073757, A083243-A083249, A083250, A083251.

Programs

  • Mathematica
    Do[r=EulerPhi[n]; d=DivisorSigma[0, n]; u=n-r-d+1; df=2*u-n; If[Equal[Abs[df], 5], Print[n(*, {d, r, u}*)]], {n, 1, 3000}]
  • PARI
    isok(n) = abs(n-2*eulerphi(n)-2*numdiv(n)+2) == 5; \\ Michel Marcus, Jul 29 2017

Extensions

a(6) from Michel Marcus, Jul 29 2017
a(7) from Amiram Eldar, Feb 02 2025

A083253 Smallest number k for which abs(A045763(k) - A073757(k)) = n, i.e., signed difference of size of related and unrelated sets to k equals either n or -n.

Original entry on oeis.org

30, 1, 2, 3, 4, 5, 8, 7, 16, 21, 32, 11, 64, 13, 84, 27, 78, 17, 200, 19, 90, 57, 140, 23, 102, 69, 120, 435, 114, 29, 132, 31, 126, 93, 392, 81, 138, 37, 156, 49, 230, 41, 168, 43, 322, 129, 260, 47, 150, 77, 180, 795, 186, 53, 204, 95, 198, 885, 280, 59, 434, 61, 228, 183
Offset: 0

Views

Author

Labos Elemer, May 07 2003

Keywords

Comments

a(258) > 10^5. - Michael De Vlieger, Jul 31 2017

Examples

			A045763(x) - A073757(x) = 0 is first satisfied at x = 30 = a(0).

Links

Crossrefs

Cf. A000005, A000010, A045763, A073757, A083243-A083249, A083250, A083251, A083252.

Programs

  • Mathematica
    With[{s = Table[Abs[n - 2 (DivisorSigma[0, n] + EulerPhi[n] - 1)], {n, 10^3}]}, TakeWhile[#, # > 0 &] &@ Flatten@ Map[FirstPosition[s, #] /. k_ /; MissingQ@ k -> 0 &, Range[0, Max@ s]]] (* Michael De Vlieger, Jul 31 2017 *)
  • PARI
    a(n) = {my(k = 1); while (abs(k - 2*(numdiv(k) + eulerphi(k) - 1)) != n, k++); k;} \\ Michel Marcus, Aug 01 2017

Formula

a(n) = min{x; abs(A045763(x) - A073757(x)) = n}.
a(p) = p, for p prime.
Showing 1-4 of 4 results.

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