A342108 Smallest positive integer m with n digits and such that omega(m) = bigomega(m) = n.

2, 10, 102, 1110, 10010, 101010, 1009470, 11741730, 223092870, 6469693230

Offset

1,1

Comments

Equivalently: smallest n-digit squarefree number with n distinct prime factors.

Differs from A036336 where length(m) = bigomega(m) = n, when length(m) is the number of digits of m (A055642) and the n prime factors of m are counted with multiplicity (A001222).

Differs from A070842 where length(m) = omega(m) = n, when length(m) is the number of digits of m (A055642) and omega(m) is the number of distinct primes factors dividing m (A001221).

The first index for which these three sequences give three distinct terms is 4:

-> a(4) = 1110 = 2 * 3 * 5 * 37 , with length(1110) = omega(1110) = bigomega(1110) = 4.

-> A036336(4) = 1012 = 2 * 2 * 11 * 23 with length(1012) = bigomega(1012) = 4 > omega(1012) = 3.

-> A070842(4) = 1020 = 2 * 2 * 3 * 5 * 17 with length(1020) = omega(1020) = 4 < bigomega(1020) = 5.

As these terms are the smallest n-digit numbers in A167050 that is finite, this sequence is also finite with 10 terms, as for A070842.

Links

Table of n, a(n) for n=1..10.

Formula

A036336(n) <= A070842(n) <= a(n).

Example

10010 = 2*5*7*11*13 is the smallest 5-digit number such that omega(10010) = bigomega(10010) = 5, hence a(5) = 10010.

Mathematica

a={}; For[n=1, n<=10, n++, For[m=10^(n-1), m<10^n, m++, If[PrimeOmega[m]==PrimeNu[m]==n, AppendTo[a, m]; Break[]]]]; a (* Stefano Spezia, Mar 04 2021 *)

Crossrefs

Subsequence of A167050.

Cf. A001221, A001222, A055642.

Cf. A036336, A070842, A342109.

Sequence in context: A036336 A070842 A086927 * A135058 A346672 A154256

Adjacent sequences: A342105 A342106 A342107 * A342109 A342110 A342111

Keyword

nonn,fini,full,base

Author

Bernard Schott, Feb 28 2021

Status

approved