A083359 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.

735, 3792, 13377, 21372, 51375, 119911, 229912, 290912, 537975, 1341275, 1713192, 2333772, 2971137, 4773132, 7747292, 13115375, 13731373, 19853575, 22940075, 29090912, 29373375, 31373137, 35322592, 52979375, 71624133, 79241575

Offset: 1

Author

Sven Simon, Apr 27 2003

Keywords

• nonn
• base

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

• 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)

Crossrefs

Cf. A083360, A083361.