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.

A083243 Numbers k for which there are more divisors and coprimes than other numbers less than k: A045763(k) < A073757(k) or A045763(k) < k/2 or A073757(k) > k/2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76
Offset: 1

Views

Author

Labos Elemer, May 07 2003

Keywords

Examples

			30 is not in the sequence because d(30) + phi(30) - 1 = 8 + 8 - 1 = 15. There are as many divisors and coprimes as there are numbers j <= 30 that neither divide nor are coprime to 30.
50 is not here because d(50) + phi(50) - 1 = 6 + 20 - 1 = 25. There are as many divisors and coprimes as there are numbers j < 50 that neither divide nor are coprime to 50.
146 is here because d(146) + phi(146) - 1 = 4 + 72 - 1 = 75; 146/2 = 73, and 75 > 73.
61455 is here because d(61455) + phi(61455) - 1 = 16 + 30720 - 1 = 30735; 61455/2 = 30727 + 1/2, and 30735 > 61455/2.

Crossrefs

Cf. A000005, A000010, A045763, A073757, A020488, A083244, A083245.

Programs

  • Mathematica
    Do[r=EulerPhi[n]; d=DivisorSigma[0, n]; u=n-r-d+1; If[Greater[n-u, n/2], Print[n, {d, r, u}]], {n, 1, 100}]
(* Second program: *)
Select[Range[120], DivisorSigma[0, #] + EulerPhi[#] - 1 > #/2 &] (* Michael De Vlieger, Aug 22 2023 *)

Formula

{ k : d(k) + phi(k) - 1 > k/2 }.

Extensions

Data corrected and entry edited by Michael De Vlieger, Aug 22 2023

A083244 k is in the sequence iff the number of numbers unrelated to k is larger than that of related ones[=divisors and coprimes] to k: A045763(k) > A073757(k) or A045763(k) > k/2 or A073757(k) < k/2.

Original entry on oeis.org

42, 54, 60, 66, 70, 72, 78, 84, 90, 96, 98, 100, 102, 108, 110, 114, 120, 126, 130, 132, 138, 140, 144, 150, 154, 156, 160, 162, 168, 170, 174, 180, 182, 186, 190, 192, 196, 198, 200, 204, 210, 216, 220, 222, 224, 228, 230, 234, 238, 240, 242, 246, 250, 252
Offset: 1

Views

Author

Labos Elemer, May 07 2003

Keywords

Examples

			k = 42 is a term because d = 8 divisors, r = 12 coprimes and u = 23 unrelated belong to it: u = 23 > 19 = 8 + 12 - 1 = d + r - 1.

Links

Crossrefs

Cf. A000005, A000010, A045763, A073757, A020488, A083243, A083245.

Programs

  • Maple
    filter:= n -> n > 2*(numtheory:-tau(n) + numtheory:-phi(n)-1):
select(filter, [$1..1000]); # Robert Israel, May 15 2017
  • Mathematica
    Do[r=EulerPhi[n]; d=DivisorSigma[0, n]; u=n-d-r+1; If[Greater[u, n/2], Print[n, {d, r, u}]], {n, 1, 100}]
(* Second program: *)
Select[Range@ 256, # - (DivisorSigma[0, #] + EulerPhi[#] - 1) > #/2 &] (* Michael De Vlieger, Jul 22 2017 *)

Formula

Numbers k such that k - d(k) - phi(k) + 1 > k/2.

A083247 Numbers k such that A000010(k) > A045763(k) > A000005(k).

Original entry on oeis.org

14, 20, 21, 22, 26, 27, 28, 32, 33, 34, 35, 38, 39, 44, 45, 46, 49, 51, 52, 55, 57, 58, 62, 63, 64, 65, 68, 69, 74, 75, 76, 77, 81, 82, 85, 86, 87, 91, 92, 93, 94, 95, 99, 106, 111, 115, 116, 117, 118, 119, 121, 122, 123, 124, 125, 128, 129, 133, 134, 135, 141, 142
Offset: 1

Views

Author

Labos Elemer, May 07 2003

Keywords

Comments

Primes are not terms since A045763(p) = 0 < A000005(p) = 2 for a prime p.

Examples

			k = 99 is a term since d(k) = 6, phi(k) = 60, unrelateds(k) = 99 - 6 - 60 + 1 = 34, and 60 > 34 > 6 holds.

Links

Crossrefs

Cf. A000005, A000010, A045763, A073757, A083243, A083244, A083245, A083246, A083248, A020488.

Programs

  • Mathematica
    Do[r=EulerPhi[n]; d=DivisorSigma[0, n]; u=n-r-d+1; If[Greater[r, u]&&Greater[u, d], Print[n, {d, r, u}]], {n, 1, 1000}]
  • PARI
    is(n)=my(r=eulerphi(n),d=numdiv(n),u=n-r-d+1); r>u && u>d \\ Charles R Greathouse IV, Feb 21 2013

A083248 Numbers k such that A045763(k) > A000010(k) > A000005(k).

Original entry on oeis.org

36, 40, 42, 48, 50, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 98, 100, 102, 104, 105, 108, 110, 112, 114, 120, 126, 130, 132, 136, 138, 140, 144, 150, 152, 154, 156, 160, 162, 168, 170, 174, 176, 180, 182, 184, 186, 190, 192, 196, 198, 200, 204, 208, 210
Offset: 1

Views

Author

Labos Elemer, May 07 2003

Keywords

Comments

Primes are not terms since A045763(p) = 0 < A000005(p) = 2 for a prime p.

Examples

			k = 100 is a term since d(k) = 9, phi(k) = 40, unrelateds(k) = 100 - 9 - 40 + 1 = 52, and 52 > 40 > 9 holds.

Links

Crossrefs

Cf. A000005, A000010, A045763, A073757, A083243, A083244, A083245, A083246, A083247, A020488.

Programs

  • Mathematica
    Do[r=EulerPhi[n]; d=DivisorSigma[0, n]; u=n-r-d+1; If[Greater[u, r]&&Greater[r, d], Print[n, {d, r, u}]], {n, 1, 1000}]
  • PARI
    isok(k) = {my(f = factor(k), d = numdiv(f), r = eulerphi(f), u = k - r - d + 1); u > r && r > d;} \\ Amiram Eldar, Feb 08 2025
Showing 1-4 of 4 results.

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