# [seqfan] Partition of multiplication

Vladimir Shevelev shevelev at bgu.ac.il
Tue Apr 30 15:57:09 CEST 2013

```Dear Seqfans,

Consider a nontrivial partition of, say, 5n, when the digits of n from the right to the left are alternatively multiplicated by 2 and by 3 (taking account carries) and after that, conversely,  alternatively multiplicated by 3 and  by 2.  The sum of the results, of course,  gives 5n. We write 5n=n<*>(2,3)+n<*>(3,2), where <*> means the above defined "mixed" multiplications. For example, 167<*>(2,3)=394, 167<*>(3,2)=441, such that 394+441=167*5=835. One can consider many new sequences
connected with mixed multiplications. For example,  put (3,2)^n=1<*>(3,2)<*>(3,2)<*>...<*>(3,2) (n times). Then
{(3,2)^n}={1,3,9,27,41,123,349,1007,...}.
Many new problems arise. For example, consider the mixed multiplications n<*>(m-2,2) with the fixed 2, denoting it
by n[*]m (m>=2). E.g., 237[*]7=237<*>(5,2)=1095. We say that 1095 is [*]-divisible by 7. Let us use Eratosthenes-like sieve over multiplication [*] to positive integers. Then what "primes" do we obtain?

Best regards,