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

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

1,4

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.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Divisor

Eric Weisstein's World of Mathematics, Permutation.

Formula

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

Example

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.

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 *)

Crossrefs

Cf. A086571, A086572.

Cf. A086957, A086958, A086959.

Cf. A086960, A086961, A086962, A086963.

Sequence in context: A142148 A142583 A327838 * A198570 A190290 A246341

Adjacent sequences: A086953 A086954 A086955 * A086957 A086958 A086959

Keyword

nonn,look

Author

Reinhard Zumkeller, Jul 25 2003

Status

approved