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

Author

Leroy Quet, Jan 13 2008

Keywords

• nonn

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)

Examples

			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).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

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

Programs

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 *)

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

Scheme

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

Formula

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

Extensions

More terms from Stefan Steinerberger, Jan 21 2008