OFFSET
1,1
COMMENTS
Larger terms can be found with the factorization of 10^m+1. A prime p containing all the prime factors of 10^m+1 will give the VFN (pp), for example 13731373 = 73*137*1373 with 73*137 = 10001. Every prime 9090...9091 builds a VFN with the cofactor 2^5.
Sequence is probably infinite.
The prime p in the 10^m+1 example above must contain exactly m digits. Also, it can contain one of the prime factors wrapped around the end of p. For example, p=11909 contains 11 and 9091, the factors of 100001, with the 9091 wrapping around to the beginning of p. This forms a(44)=1190911909. - Deron Stewart, Feb 23 2019
The concatenation must be possible using the prime factors of the number, unlimited multiplicity of the distinct prime factors is not allowed. For example, 71153775 = 3*3*3*5*5*7*11*37*37 can be formed by 7||11||5||37||7||5 but the concatenation requires two 7's and there is only one 7 in the prime factorization, so it is not in the sequence. - Deron Stewart, Mar 01 2019
REFERENCES
J. A. Lindon, Visible factor numbers, J. Rec. Math., 1 (1968), 217.
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..165 (first 64 terms except a(59) from Sven Simon, a(59) and a(64)-a(74) from Deron Stewart)
EXAMPLE
From Sean A. Irvine, Jan 20 2026: (Start)
11913 is not a term because although 11913 = 3*11*19 contains each of 3, 11, and 19, we cannot cover the third 1:
11913
11
19
3. [Based on personal communication from Sven Simon] (End)
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Sven Simon, Apr 27 2003
EXTENSIONS
Name clarified by Sean A. Irvine, Jan 20 2026
STATUS
approved