A046461 Numbers k such that concatenation of numbers from 1 to k is a semiprime.

3, 4, 7, 34, 97

Offset: 1

Author

Patrick De Geest, Aug 15 1998

Keywords

• nonn
• hard
• base
• more

Comments

From Sean A. Irvine, Apr 15 2010, updated Oct 08 2015: (Start)

5053 and 9706 are definite terms of the sequence.

The next potential term is 1651.

A007908(1651) is composite, but has no known prime factor, and its least prime factor likely has at least 45 digits. (End)

If k is a multiple of 10, then k is not a term. - Chai Wah Wu, Jan 22 2020

From Jon E. Schoenfield, Oct 07 2023: (Start)

k cannot be a term if any of the following are true:

4|k and k > 4 (2*2 would divide the concatenation)

6|k or 6|k-2 (2*3 " " " " )

9|k or 9|k-8 (3*3 " " " " )

10|k (2*5 " " " " )

15|k or 15|k-5 (3*5 " " " " )

25|k (5*5 " " " " ) (End)

Examples

			A007908(691)=1304238680165623831238651513722972177904593843651*C1916, so A007908(691) is not a semiprime and 691 is not a term of this sequence.
		

Links

• Patrick De Geest, Normal Smarandache Concatenated Numbers, Prime factors from 1 up to n.
• M. Fleuren, Factors and primes of Smarandache sequences.
• M. Fleuren, Smarandache Factors and Reverse factors.
• Carlos Rivera, Puzzle 8. Primes by Listing, The Prime Puzzles and Problems Connection.

Crossrefs

Cf. A007908, A046460.

Programs

Mathematica

Select[Range[100], Length@FactorInteger@FromDigits@Flatten@IntegerDigits@Range@# == 2 &] (* Robert Price, Oct 11 2019 *)
Select[Range[100],PrimeOmega[FromDigits[Flatten[IntegerDigits/@Range[#]]]] == 2&] (* Harvey P. Dale, Sep 10 2022 *)

Extensions

Simplified definition by Sean A. Irvine, Mar 29 2010

a(5) from Sean A. Irvine, Mar 29 2010