A342109 -id:A342109 - cp's OEIS frontend

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

Bernard Schott, Feb 28 2021

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 prime 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.

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

Subsequence of A167050.
Cf. A001221, A001222, A055642.
Cf. A036336, A070842, A342109.

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 *)

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

cp's OEIS Frontend

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

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