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A086956
a(1)=1, for n>1: a(n) is the smallest divisor of n occurring earlier at most twice.
10
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
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
KEYWORD
nonn,look
AUTHOR
Reinhard Zumkeller, Jul 25 2003
STATUS
approved