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 1-9 of 9 results.

A086572 Solution to A086571(a(n))=A086571(n) and a(n)<>n.

Original entry on oeis.org

2, 1, 9, 6, 10, 4, 14, 12, 3, 5, 22, 8, 26, 7, 45, 32, 34, 24, 38, 30, 63, 11, 46, 18, 50, 13, 36, 42, 58, 20, 62, 16, 99, 17, 70, 27, 74, 19, 117, 100, 82, 28, 86, 66, 15, 23, 94, 60, 98, 25, 153, 78, 106, 72, 110, 84, 171, 29, 118, 48, 122, 31, 21, 80, 130, 44, 134
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 22 2003

Keywords

Comments

Well defined as each number occurs exactly twice in A086571;
a self-inverse permutation of natural numbers: a(a(n))=n.
A086938(n) = a(A086939(n)), A086939(n) = a(A086938(n)). - Reinhard Zumkeller, Jul 24 2003

Examples

			A086571(8)=4 and A086571(12)=4, therefore a(8)=12 and a(12)=8.

Links

A086938 Smallest m with A086571(m) = n.

Original entry on oeis.org

1, 4, 3, 8, 5, 18, 7, 16, 27, 20, 11, 48, 13, 28, 15, 64, 17, 54, 19, 40, 21, 44, 23, 96, 25, 52, 81, 56, 29, 90, 31, 128, 33, 68, 35, 144, 37, 76, 39, 200, 41, 126, 43, 88, 135, 92, 47, 192, 49, 250, 51, 104, 53, 162, 55, 112, 57, 116, 59, 360, 61, 124, 189, 256
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 24 2003

Keywords

Comments

a(n) < A086939(n); A086571(a(n)) = A086571(A086939(n)) = n;
n and A086571 are divisors of a(n);
a(n) = A086572(A086939(n)), A086939(n) = A086572(a(n)).

Links

A086939 Greatest m with A086571(m) = n.

Original entry on oeis.org

2, 6, 9, 12, 10, 24, 14, 32, 36, 30, 22, 60, 26, 42, 45, 80, 34, 72, 38, 100, 63, 66, 46, 120, 50, 78, 108, 84, 58, 150, 62, 160, 99, 102, 70, 180, 74, 114, 117, 240, 82, 168, 86, 132, 225, 138, 94, 288, 98, 300, 153, 156, 106, 216, 110, 224, 171, 174, 118
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 24 2003

Keywords

Comments

A086938(n) < a(n); A086571(a(n)) = A086571(A086938(n)) = n;
n and A086571 are divisors of a(n);
a(n) = A086572(A086938(n)), A086938(n) = A086572(a(n)).

Links

A086956 a(1)=1, for n>1: a(n) is the smallest divisor of n occurring earlier at most twice.

Original entry on oeis.org

1, 1, 1, 2, 5, 2, 7, 2, 3, 5, 11, 3, 13, 7, 3, 4, 17, 6, 19, 4, 7, 11, 23, 4, 5, 13, 9, 14, 29, 6, 31, 8, 11, 17, 35, 6, 37, 19, 13, 8, 41, 14, 43, 22, 9, 23, 47, 8, 49, 10, 17, 26, 53, 9, 55, 14, 19, 29, 59, 10, 61, 31, 21, 16, 65, 22, 67, 34, 23, 10, 71, 12, 73, 37, 15, 38
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 25 2003

Keywords

Comments

For all natural numbers m there exist exactly three numbers u(m) < v(m) < w(m) with m=a(u(m))=a(v(m))=a(w(m)) (see A086957=u, A086958=v, A086959=w).
Permuting {u,v,w} induces 6=3! permutations of natural numbers: [(2,3,1)]->A086960, [(3,2,1)]->A086961, [(1)(2,3)]->A086962, [(2)(3,1)]->A086963, [(3)(2,1)]->A086964 and [(1,2,3)]->A000027.

Examples

			Divisor set of n=20: {1,2,4,5,10,20},
divisors occurring < 20: 1=a(1)=a(2)=a(3), 2=a(4)=a(6)=a(8), 4=a(16),
and as 4 occurs only once a(20)=4.

Links

Crossrefs

Cf. A086571, A086572.
Cf. A086957, A086958, A086959.
Cf. A086960, A086961, A086962, A086963.

Programs

  • Maple
    N:= 100:
V:= Vector(N):
for n from 1 to N do
  Dn:= select(t -> V[t]<=2,numtheory:-divisors(n));
  v:= min(Dn);
  A[n]:= v; V[v]:= V[v]+1
od:
seq(A[i],i=1..N); # Robert Israel, Aug 01 2019
  • Mathematica
    nn = 100;
V = Table[0, {nn}];
For[n = 1, n <= nn, n++,
     Dn = Select[Divisors[n], V[[#]] <= 2&];
     v = Min[Dn];
     a[n] = v; V[[v]] = V[[v]]+1];
Table[a[n], {n, 1, nn}] (* Jean-François Alcover, Dec 13 2021, after Robert Israel *)

Formula

a(p) = p for primes p>3.

A086962 a(A086957(n))=A086957(n), a(A086958(n))=A086959(n), a(A086959(n))=A086958(n), see A086956.

Original entry on oeis.org

1, 3, 2, 4, 5, 8, 7, 6, 9, 25, 11, 15, 13, 21, 12, 16, 17, 18, 19, 24, 14, 33, 23, 20, 10, 39, 27, 28, 29, 36, 31, 32, 22, 51, 35, 30, 37, 57, 26, 48, 41, 56, 43, 44, 54, 69, 47, 40, 49, 50, 34, 52, 53, 45, 55, 42, 38, 87, 59, 70, 61, 93, 63, 64, 65, 88, 67, 68, 46
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 25 2003

Keywords

Comments

A self-inverse permutation of natural numbers: a(a(n))=n;
a(n) = A086962(A086963(n)) = A086963(A086962(n)).

Links

Crossrefs

Cf. A086572, A086571.

A086963 a(A086957(n))=A086959(n), a(A086958(n))=A086958(n), a(A086959(n))=A086957(n), see A086956.

Original entry on oeis.org

3, 2, 1, 8, 25, 6, 21, 4, 15, 10, 33, 12, 39, 14, 9, 24, 51, 36, 57, 20, 7, 22, 69, 16, 5, 26, 54, 56, 87, 30, 93, 48, 11, 34, 280, 18, 111, 38, 13, 40, 123, 42, 129, 88, 45, 46, 141, 32, 343, 70, 17, 104, 159, 27, 275, 28, 19, 58, 177, 60, 183, 62, 168, 112, 325
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 25 2003

Keywords

Comments

A self-inverse permutation of natural numbers: a(a(n))=n;
a(n) = A086962(A086963(n)) = A086963(A086962(n)).

Links

Crossrefs

Cf. A086572, A086571.

A086960 a(A086957(n))=A086958(n), a(A086958(n))=A086959(n), a(A086959(n))=A086957(n), see A086956.

Original entry on oeis.org

2, 3, 1, 6, 10, 8, 14, 4, 12, 25, 22, 15, 26, 21, 9, 20, 34, 30, 38, 24, 7, 33, 46, 16, 5, 39, 45, 42, 58, 36, 62, 40, 11, 51, 245, 18, 74, 57, 13, 48, 82, 56, 86, 66, 54, 69, 94, 32, 98, 60, 17, 78, 106, 27, 110, 28, 19, 87, 118, 70, 122, 93, 147, 80, 130, 88, 134
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 25 2003

Keywords

Comments

permutation of natural numbers with inverse A086961: a(A086961(n)) = A086961(a(n)) = n;
a(n) = A086962(A086963(n)) = A086963(A086962(n)).

Links

Crossrefs

Cf. A086572, A086571.

A086961 a(A086957(n))=A086959(n), a(A086958(n))=A086957(n), a(A086959(n))=A086958(n), see A086956.

Original entry on oeis.org

3, 1, 2, 8, 25, 4, 21, 6, 15, 5, 33, 9, 39, 7, 12, 24, 51, 36, 57, 16, 14, 11, 69, 20, 10, 13, 54, 56, 87, 18, 93, 48, 22, 17, 280, 30, 111, 19, 26, 32, 123, 28, 129, 88, 27, 23, 141, 40, 343, 70, 34, 104, 159, 45, 275, 42, 38, 29, 177, 50, 183, 31, 168, 112, 325
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 25 2003

Keywords

Comments

permutation of natural numbers with inverse A086960: a(A086960(n)) = A086960(a(n)) = n;
a(n) = A086962(A086963(n)) = A086963(A086962(n)).

Links

Crossrefs

Cf. A086572, A086571.

A086964 a(A086957(n))=A086958(n), a(A086958(n))=A086957(n), a(A086959(n))=A086959(n), see A086956.

Original entry on oeis.org

2, 1, 3, 6, 10, 4, 14, 8, 12, 5, 22, 9, 26, 7, 15, 20, 34, 30, 38, 16, 21, 11, 46, 24, 25, 13, 45, 42, 58, 18, 62, 40, 33, 17, 245, 36, 74, 19, 39, 32, 82, 28, 86, 66, 27, 23, 94, 48, 98, 60, 51, 78, 106, 54, 110, 56, 57, 29, 118, 50, 122, 31, 147, 80, 130, 44, 134
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 25 2003

Keywords

Comments

A self-inverse permutation of natural numbers: a(a(n))=n;
a(n) = A086962(A086963(n)) = A086963(A086962(n)).

Links

Crossrefs

Cf. A086572, A086571.
Showing 1-9 of 9 results.

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

Content is available under The OEIS End-User License Agreement.