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

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, 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, 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

Offset

1,2

Comments

From Amiram Eldar, Jun 30 2025: (Start)

The first position of k = 1, 2, 3, ... is A006939(k-1).

Let d(k) be the asymptotic density of the occurrences of k = 1, 2, ... in this sequence.

d(1) = 0 = the density of the powerful numbers (A001694).

d(2) = Product_{primes p} (1 - 1/p^2 + 1/p^3) = 0.748535... (A330596) = the density of A337050.

d(3) = Product_{primes p} (1 - 1/p^3 + 1/p^4) - Product_{primes p} (1 - 1/p^2 + 1/p^4) = 0.23548870893364493209...

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... .

d(5) = 0.000174471282..., d(6) = 0.000000217516..., etc.

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)).

The asymptotic mean of this sequence is Sum_{k>=1} k*d(k) = 2.26761567808299143335... . (End)

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from exponents in factorization of n.

Formula

a(n) = A257993(A181819(n)). - Antti Karttunen, Jul 28 2017

Example

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).

Mathematica

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 *)

Prog

(Scheme) (define (A134193 n) (A257993 (A181819 n))) ;; Antti Karttunen, Jul 28 2017
(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

Crossrefs

Cf. A001694, A006939, A136567, A181819, A257993, A330596, A337050.

Sequence in context: A072463 A128853 A136165 * A230259 A085030 A078377

Adjacent sequences: A134190 A134191 A134192 * A134194 A134195 A134196

Keyword

nonn

Author

Leroy Quet, Jan 13 2008

Extensions

More terms from Stefan Steinerberger, Jan 21 2008

Status

approved