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A178442 Two numbers k and l we call equivalent if they have the same vector of exponents with positive components in prime power factorization. Let a(1)=1, a(2)=3. Then a(n)>a(n-1) is the smallest number equivalent to n.

%I A178442 #17 Jul 03 2020 20:00:44
%S A178442 1,3,5,9,11,14,17,27,49,51,53,63,67,69,74,81,83,98,101,116,118,119,
%T A178442 127,135,169,177,343,356,359,366,367,3125,3127,3131,3133,3249,3251,
%U A178442 3254,3261,3272,3299,3302,3307,3308,3316,3317,3319,3321,3481
%N A178442 Two numbers k and l we call equivalent if they have the same vector of exponents with positive components in prime power factorization. Let a(1)=1, a(2)=3. Then a(n)>a(n-1) is the smallest number equivalent to n.
%C A178442 Note that, e.g., 12 and 50 have similar structure in their prime power factorizations, but are not equivalent: their vectors of exponents are (2,1) and (1,2). On the other hand, 6 and 35 are equivalent with the same vector (1,1).
%C A178442 Question. What is the growth of the sequence?
%H A178442 Harvey P. Dale, <a href="/A178442/b178442.txt">Table of n, a(n) for n = 1..1000</a>
%t A178442 nxt[{n_,a_}]:=Module[{j=FactorInteger[n+1][[All,2]],k=a+1},While[ j!= FactorInteger[k][[All,2]],k++];{n+1,k}]; Join[{1},NestList[nxt,{2,3},50][[All,2]]] (* _Harvey P. Dale_, Jul 03 2020 *)
%o A178442 (Sage)
%o A178442 prime_signature = lambda n: [m for p, m in factor(n)]
%o A178442 @CachedFunction
%o A178442 def A178442(n):
%o A178442     if n <= 2: return {1:1, 2:3}[n]
%o A178442     psig_n = prime_signature(n)
%o A178442     return next(k for k in IntegerRange(A178442(n-1)+1,infinity) if prime_signature(k) == psig_n)
%o A178442 # _D. S. McNeil_, Dec 22 2010
%Y A178442 Cf. A172980, A172999
%K A178442 nonn
%O A178442 1,2
%A A178442 _Vladimir Shevelev_, Dec 22 2010
%E A178442 Corrected and extended by _D. S. McNeil_, Dec 22 2010