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 1-3 of 3 results.

A083245 Difference between numbers of related and numbers of unrelated numbers belonging to n: a(n) = A073757(n)-A045763(n) = (n-u(n))-u(n) = n-2*A045763(n) = 2*A073757(n)-n.

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 7, 6, 7, 4, 11, 6, 13, 4, 7, 8, 17, 4, 19, 6, 9, 4, 23, 6, 19, 4, 15, 6, 29, 0, 31, 10, 13, 4, 19, 4, 37, 4, 15, 6, 41, -4, 43, 6, 13, 4, 47, 2, 39, 0, 19, 6, 53, -4, 31, 6, 21, 4, 59, -6, 61, 4, 19, 12, 37, -12, 67, 6, 25, -8, 71, -2, 73, 4, 15, 6, 49, -16, 79, 2, 35, 4, 83, -14, 49, 4, 31, 6, 89, -20, 59, 6, 33, 4, 55, -10, 97, -4
Offset: 1

Views

Author

Labos Elemer, May 07 2003

Keywords

Comments

There are only 2 cases [n=30, n=50] below 10^7 such that a(n) = 0.
No other zeros found up to 10^9. - Michel Marcus, Jul 30 2017

Examples

			n=37, d=2,r=36,u=0, a(37)=2+36-1-0=37>0; primes are fixed points.
n=42, d=8,r=12,u=23,a(42)=8+12-1-23=-4<0, terms of A083244;
n=30, d=8,r=8,u=15, a(30)=0;
n=50, d=6,r=20,u=25,a(50)=0.

Links

Crossrefs

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

Programs

  • Mathematica
    Table[2*(DivisorSigma[0, w]+EulerPhi[w]-1)-w, {w, 1, 1000}]
  • PARI
    a(n) = 2*(numdiv(n)+eulerphi(n)-1) - n; \\ Michel Marcus, Jul 30 2017

Formula

a(n) = 2(A000005(n)+A000010(n)-1)-n.

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 1-3 of 3 results.

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