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A371473 a(1) = 1; for n>1, if a(n-1) is squarefree, a(n) = a(n-1) + n, otherwise a(n) = squarefree kernel of a(n-1).

%I A371473 #31 Apr 20 2024 10:14:27
%S A371473 1,3,6,10,15,21,28,14,23,33,44,22,35,49,7,23,40,10,29,49,7,29,52,26,
%T A371473 51,77,104,26,55,85,116,58,91,125,5,41,78,116,58,98,14,56,14,58,103,
%U A371473 149,196,14,63,21,72,6,59,113,168,42,99,33,92,46
%N A371473 a(1) = 1; for n>1, if a(n-1) is squarefree, a(n) = a(n-1) + n, otherwise a(n) = squarefree kernel of a(n-1).
%C A371473 Inspired by Recaman's sequence A005132.
%C A371473 Some nonsquarefree numbers will not appear in this sequence. However, I conjecture that all squarefree numbers will appear. First occurrence of 2 is at a(766) = 2.
%H A371473 Joseph C. Y. Wong, <a href="/A371473/b371473.txt">Table of n, a(n) for n = 1..10000</a>
%e A371473 a(1) = 1 is squarefree, so a(2) = a(1) + 2 = 3.
%e A371473 a(7) = 28 = 2*2*7 is not squarefree, so a(8) = 2*7 = 14.
%t A371473 rad[n_]:=Product[Part[First/@FactorInteger[n],i],{i,Length[FactorInteger[n]]}]; a[1]=1; a[n_]:=If[SquareFreeQ[a[n-1]],a[n-1]+n,rad[a[n-1]]]; Array[a,60] (* _Stefano Spezia_, Mar 26 2024 *)
%o A371473 (Python)
%o A371473 from numpy import prod
%o A371473 def primefact(a):
%o A371473   factors = []
%o A371473   d = 2
%o A371473   while a > 1:
%o A371473     while a % d == 0:
%o A371473       factors.append(d)
%o A371473       a /= d
%o A371473     d = d + 1
%o A371473   return factors
%o A371473 def squarefree(a):
%o A371473   return sorted(list(set(primefact(a)))) == sorted(primefact(a))
%o A371473 sequence = [1]
%o A371473 a = 1
%o A371473 for n in range(1, 1001):
%o A371473   if not squarefree(a):
%o A371473     a = prod(list(set(primefact(a))))
%o A371473   else:
%o A371473     a += n+1
%o A371473   sequence.append(a)
%o A371473 print(sequence)
%o A371473 (PARI) lista(nn) = my(v = vector(nn)); v[1] = 1; for (n=2, nn, if (issquarefree(v[n-1]), v[n] = v[n-1]+n, v[n] = factorback(factor(v[n-1])[,1]));); v; \\ _Michel Marcus_, Mar 26 2024
%Y A371473 Cf. A005132, A005117, A007947.
%K A371473 nonn
%O A371473 1,2
%A A371473 _Joseph C. Y. Wong_, Mar 24 2024