A324261 Subsequence of A083359 (Visual Factor Numbers) of the form (10^m+1)*p, where the decimal representation of prime p contains all the prime factors of 10^m+1.

13731373, 31373137, 1190911909, 9091190911, 19090911909091, 7316763215964848081373167632159648480813, 111272689909091345969111272689909091345969, 111279090913268945969111279090913268945969, 112726894596919090913112726894596919090913

Offset

1,1

Comments

Subsequence of A020338 (Doublets: base-10 representation is the juxtaposition of two identical strings).

The prime factors of 10^m + 1 are also prime factors of 10^km + 1, where k is odd, so a(n) and a(km) have those prime factors in common.

Expanding on the comment in A083359 regarding finding large terms, we can generate large doublet terms in the following way:

-- for any composite number M = 10^m + 1 try to compose a prime number p of length m from the prime factors of M. Generally this will require the factors to overlap to reduce the length to m.

-- the factors can wrap around to the beginning of p. For example, if M has a factor of 137 then p can be of the form 7...13.

-- the term is formed by concatenating p with itself to form a(n) = p||p. The resulting number will consist entirely of the concatenation of its prime factors with allowed overlap as in A083359.

Links

Table of n, a(n) for n=1..9.

Example

With m = 4: 10^4 + 1 = 10001 = 73 * 137. We can form prime p = 1373 which concatenates with itself to give a(1) = 13731373 = 73 * 137 * 1373. We can also form the prime p = 3137 which gives a(2). The number 7313 also contains all the prime factors of 10001 but it is not prime.
With m = 33: 10^33 + 1 = 7*11*11*13*23*4093*8779*599144041*183411838171, there are 4932 m-digit numbers that contain all the factors, of which 227 of them are prime. Each of these primes generates a term in the sequence with 66 digits, the smallest of which is 112359914404134093877918341183817112359914404134093877918341183817. This is A324262(33).

Crossrefs

Cf. A083359, A020338, A324262, A324257.

Sequence in context: A126724 A251565 A082256 * A324262 A321364 A116148

Adjacent sequences: A324258 A324259 A324260 * A324262 A324263 A324264

Keyword

nonn,base

Author

Deron Stewart, Mar 13 2019

Status

approved