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A134193 a(1) = 1; for n>1, a(n) = the smallest positive integer not occurring among the exponents in the prime-factorization of n.

%I A134193 #21 Jun 30 2025 04:24:57
%S A134193 1,2,2,1,2,2,2,1,1,2,2,3,2,2,2,1,2,3,2,3,2,2,2,2,1,2,1,3,2,2,2,1,2,2,
%T A134193 2,1,2,2,2,2,2,2,2,3,3,2,2,2,1,3,2,3,2,2,2,2,2,2,2,3,2,2,3,1,2,2,2,3,
%U A134193 2,2,2,1,2,2,3,3,2,2,2,2,1,2,2,3,2,2,2,2,2,3,2,3,2,2,2,2,2,3,3,1,2,2,2,2,2,2
%N A134193 a(1) = 1; for n>1, a(n) = the smallest positive integer not occurring among the exponents in the prime-factorization of n.
%C A134193 From _Amiram Eldar_, Jun 30 2025: (Start)
%C A134193 The first position of k = 1, 2, 3, ... is A006939(k-1).
%C A134193 Let d(k) be the asymptotic density of the occurrences of k = 1, 2, ... in this sequence.
%C A134193 d(1) = 0 = the density of the powerful numbers (A001694).
%C A134193 d(2) = Product_{primes p} (1 - 1/p^2 + 1/p^3) = 0.748535... (A330596) = the density of A337050.
%C A134193 d(3) = Product_{primes p} (1 - 1/p^3 + 1/p^4) - Product_{primes p} (1 - 1/p^2 + 1/p^4) = 0.23548870893364493209...
%C A134193 d(4) = Product_{primes p} (1 - 1/p^4 + 1/p^5) - Product_{primes p} (1 - 1/p^3 + 1/p^5) - Product_{primes p} (1 - 1/p^2 + 1/p^3 - 1/p^4 + 1/p^5) + Product_{primes p} (1 - 1/p^2 + 1/p^5) = 0.01580134256336122613... .
%C A134193 d(5) = 0.000174471282..., d(6) = 0.000000217516..., etc.
%C A134193 In general, d(k) = Sum_{s subset of {2, 3, ..., k-1}} (-1)^card(s) * Product_{p prime} (1 -Sum_{i=1..card(s)} 1/p^s(i) + 1/p^(s(i)+1) - 1/p^k + 1/p^(k+1)).
%C A134193 The asymptotic mean of this sequence is Sum_{k>=1} k*d(k) = 2.26761567808299143335... . (End)
%H A134193 Antti Karttunen, <a href="/A134193/b134193.txt">Table of n, a(n) for n = 1..10000</a>
%H A134193 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F A134193 a(n) = A257993(A181819(n)). - _Antti Karttunen_, Jul 28 2017
%e A134193 The prime factorization of 24 is 2^3 * 3^1. The exponents are 3 and 1. Therefore a(24) = 2 is the smallest positive integer not occurring among (3,1).
%t A134193 Join[{1}, Table[Complement[Range[n], Table[FactorInteger[n][[i, 2]], {i, 1, Length[FactorInteger[n]]}]][[1]], {n, 2, 120}]] (* _Stefan Steinerberger_, Jan 21 2008 *)
%o A134193 (Scheme) (define (A134193 n) (A257993 (A181819 n))) ;; _Antti Karttunen_, Jul 28 2017
%o A134193 (PARI) a(n) = if (n==1, 1, my(f=factor(n)); ve = vecsort(f[,2],,8); k = 1; while(vecsearch(ve, k), k++); k;); \\ _Michel Marcus_, Jul 28 2017
%Y A134193 Cf. A001694, A006939, A136567, A181819, A257993, A330596, A337050.
%K A134193 nonn
%O A134193 1,2
%A A134193 _Leroy Quet_, Jan 13 2008
%E A134193 More terms from _Stefan Steinerberger_, Jan 21 2008