A125315 Smallest n-digit number that has exactly n divisors, each with a different number of digits, or 0 if no such number exists.

1, 11, 121, 1111, 14641, 112211, 1771561, 11117777, 123187801, 1144664411, 25937424601, 111255594439, 3138428376721, 11676721656611, 125415159629881, 1111777777824439, 45949729863572161

Offset

1,2

Comments

From the 2006-07 Mandelbrot competition by Sam Vandervelde, which asked for the smallest composite number in this collection.

There is no 29-digit number with this property, because to have 29 factors it must be of the form p^28, but no number of that form has 29 digits.

Comments from Farideh Firoozbakht and David W. Wilson, Dec 14 2006: (Start)

"If p is a prime greater than 23 then a(p) = 0. Proof. Suppose a(p) = M > 0. If p is prime then M must be a p-digit number of the form q^(p-1) where q is prime. But if q <= 7 the number of digits of q^(p-1) is less than p and if q > 7 & p > 23 the number of digits of q^(p-1) is greater than p. Hence if p is a prime greater than 23, M doesn't exist.

"But for many numbers n greater than 29, a(n) > 0. For example 10^53999 < a(54000) <= 11^2*1009^5*(10^18+9)^2999. Proof : if n = 11^2*1009^5*(10^18+9)^2999 then n has exactly 54000 divisors d_k (k=1,2, ..., 54000) and each d_k has exactly k digits. Hence a(54000) exists and a(54000) is a 54000-digit number less than n+1.

"In fact if 0 < m <= 3000 then a(18m) exists and a(18m) <= 11 * 101^2 * 1000003^2 * (10^18+9)^(m-1). The right hand side is an 18m-digit number for 1 <= m <= 243040916832487184.

"On the other hand, under generous assumptions about the size of prime gaps, we have a(2^m) <= Product_{0 <= k < m} nextPrime(10^(2^k)), where the right side has 2^m digits, which would provide an infinitude of numbers with precisely one divisor of every possible length." (End)

Further comments from Farideh Firoozbakht, Dec 17 2006: Perhaps for each natural number m we have a(2^m) = Product_{0 <= k < m} NextPrime(10^(2^k)), namely a(2^m) = a(2^(m-1))* NextPrime(10^(2^(m-1))). This would give a(2^1) = 11, a(2^2) = 1111 = 11*101, a(2^3) = 11117777 = 11*101*10007, a(2^4) = 1111777777824439 = 11*101*10007*100000007, a(2^5) = 11117777778244457818444447290779 =11*101*10007*100000007* 10000000000000061.

Links

David W. Wilson, Table of n, a(n) for n = 1..29

Example

1: 1
11: 1 11
121: 1 11 121
1111: 1 11 101 1111

Crossrefs

See A125845 for the list of all numbers with this property.

Sequence in context: A033867 A307861 A106473 * A223676 A132583 A007907

Adjacent sequences: A125312 A125313 A125314 * A125316 A125317 A125318

Keyword

nonn,base

Author

Joshua Zucker, Dec 11 2006

Extensions

More terms from David W. Wilson, Dec 11 2006

Edited by N. J. A. Sloane, Dec 22 2006

Status

approved