A083359 - OEIS

%I #27 Feb 28 2021 00:45:43

%S 735,3792,13377,21372,51375,119911,229912,290912,537975,1341275,

%T 1713192,2333772,2971137,4773132,7747292,13115375,13731373,19853575,

%U 22940075,29090912,29373375,31373137,35322592,52979375,71624133,79241575

%N Visible Factor Numbers, or VFNs: numbers n with the property that every prime factor of n can be found in the decimal expansion of n and every digit of n can be found in a prime factor.

%C 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.

%C Sequence is probably infinite.

%C 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

%C 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

%D Lindon, Visible factor numbers, J. Rec. Math., 1 (1968), 217.

%H Giovanni Resta, <a href="/A083359/b083359.txt">Table of n, a(n) for n = 1..165</a> (first 64 terms except a(59) from Sven Simon, a(59) and a(64)-a(74) from Deron Stewart)

%Y Cf. A083360, A083361.

%K nonn,base

%O 1,1

%A _Sven Simon_, Apr 27 2003