Multiplicative sequences

[Antti Karttunen]

"David W. Wilson" wrote:

> ------------------------------------------------------------------------
> I am thinking of downloading the EIS and trying to identify multiplicative
> sequences and clean them up a bit.

Should there exists also a keywoard "expo" for "exponential sequences"?

I.e. sequences like A000079 (Powers of 2) are unconditionally exponential:
a(i+j) = a(i)*a(j)

and sequences like

A001317 1,3,5,15,17,51,85,255,257,771,1285,3855,4369,13107,21845,
(Pascal's triangle mod 2 converted to decimal, a(0) = 1, a(1) = 3)

and A050613 1,1,3,3,7,7,21,21,47,47,141,141,329,329,987,987,2207,2207,
                      6621,6621,15449,15449,46347,46347,103729,103729,311187,
                      311187,726103,726103,2178309,2178309,4870847,4870847,
                      14612541,14612541,34095929,34095929
(Products of distinct terms of 1 and rest from A001566: a(n) =
PRODUCT(L(2^i)^bit(n,i),i=0..[log2(n+1)])).

are conditionally exponential, the condition being here, that the indices i & j do
not have common 1-bits
in their binary expansion:
a(i+j) = a(i)*a(j), if AND(i,j) = 0.

Compare this to usual condition on multiplicative sequences, that the indices
i & j have no common "non-zero position" in their prime factorizations.

Other kind of meaningful conditions could be found as well.


Terveisin,

Antti Karttunen