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

Author

Reinhard Zumkeller, Jul 25 2003

Keywords

• nonn
• look

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

• 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.

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.