cp's OEIS Frontend

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

%I A086956 #17 Feb 16 2025 08:32:50
%S A086956 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,
%T A086956 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,
%U A086956 14,19,29,59,10,61,31,21,16,65,22,67,34,23,10,71,12,73,37,15,38
%N A086956 a(1)=1, for n>1: a(n) is the smallest divisor of n occurring earlier at most twice.
%C A086956 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).
%C A086956 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.
%H A086956 Robert Israel, <a href="/A086956/b086956.txt">Table of n, a(n) for n = 1..10000</a>
%H A086956 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Divisor.html">Divisor</a>
%H A086956 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Permutation.html">Permutation.</a>
%F A086956 a(p) = p for primes p>3.
%e A086956 Divisor set of n=20: {1,2,4,5,10,20},
%e A086956 divisors occurring < 20: 1=a(1)=a(2)=a(3), 2=a(4)=a(6)=a(8), 4=a(16),
%e A086956 and as 4 occurs only once a(20)=4.
%p A086956 N:= 100:
%p A086956 V:= Vector(N):
%p A086956 for n from 1 to N do
%p A086956   Dn:= select(t -> V[t]<=2,numtheory:-divisors(n));
%p A086956   v:= min(Dn);
%p A086956   A[n]:= v; V[v]:= V[v]+1
%p A086956 od:
%p A086956 seq(A[i],i=1..N); # _Robert Israel_, Aug 01 2019
%t A086956 nn = 100;
%t A086956 V = Table[0, {nn}];
%t A086956 For[n = 1, n <= nn, n++,
%t A086956      Dn = Select[Divisors[n], V[[#]] <= 2&];
%t A086956      v = Min[Dn];
%t A086956      a[n] = v; V[[v]] = V[[v]]+1];
%t A086956 Table[a[n], {n, 1, nn}] (* _Jean-François Alcover_, Dec 13 2021, after _Robert Israel_ *)
%Y A086956 Cf. A086571, A086572.
%Y A086956 Cf. A086957, A086958, A086959.
%Y A086956 Cf. A086960, A086961, A086962, A086963.
%K A086956 nonn,look
%O A086956 1,4
%A A086956 _Reinhard Zumkeller_, Jul 25 2003