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

1,1

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

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.

Sequence in context: A227755 A096595 A324257 * A324258 A083360 A324260

Adjacent sequences: A083356 A083357 A083358 * A083360 A083361 A083362

Keyword

nonn,base

Author

Sven Simon, Apr 27 2003

Status

approved