Additive sequences

[franktaw at netscape.net]

There was some discussion on this list a while back
about what to call sequences where
a(n*m) = a(n) + a(m) whenever gcd(n,m) = 1.
I don't think there was any consensus reached,
but one of the possibilities, "additive", was and is
in use in the database, so I guess it wins by default.

In any event, here are some lists of additive
sequences.  (Totally additive means that
a(n*m) = a(n) + a(m) unconditionally.)  These
are supplied with the idea that index entries
could be made for them.

I'm posting these here instead of just sending
them to Neil in the hopes that others will
identify additional additive sequences.

Franklin T. Adams-Watters

[Anyone replying to this, please cut here.]

Totally additive:

A001222, A001414, A007814, A007949, A048675, A056239, A067666, A076649, 
A078458, A078908, A078909, A112765, A113177.

Additive but not totally additive:

A001221, A005063-A005085, A005087-A005091, A005094, A008472, A008474, 
A008475, A008476, A046660, A052331, A055631, A056169, A056170, A059841, 
A064372, A064415, A066328, A079978, A080256, A081403, A087207, A090885, 
A106490, A106492, A113178, A113222, A115357, A121262.

Totally additive fractions:

A083345/A083346.

Additive but not totally additive fractions:

A028235/A007947, A028236/A000027.

And, mostly for Neil, here are edits to the sequences (new lines only),
to describe these sequences (with a couple of other minor edits
thrown in.

%F A001221 Additive with a(p^e) = 1.
%C A001222 Number of prime powers (not including 1) that divide n.
%F A001222 Totally additive with a(p) = 1.
%F A005063 Additive with a(p^e) = p^2.
%F A005064 Additive with a(p^e) = p^3.
%F A005065 Additive with a(p^e) = p^4.
%F A005066 Additive with a(p^e) = 0 if p = 2, p^2 otherwise.
%F A005067 Additive with a(p^e) = 0 if p = 2, p^3 otherwise.
%F A005068 Additive with a(p^e) = 0 if p = 2, p^4 otherwise.
%F A005069 Additive with a(p^e) = 0 if p = 2, p otherwise.
%F A005070 Additive with a(p^e) = p if p = 1 (mod 3), 0 otherwise.
%F A005071 Additive with a(p^e) = p^2 if p = 1 (mod 3), 0 otherwise.
%F A005072 Additive with a(p^e) = p^3 if p = 1 (mod 3), 0 otherwise.
%F A005073 Additive with a(p^e) = p^4 if p = 1 (mod 3), 0 otherwise.
%F A005074 Additive with a(p^e) = p if p = 2 (mod 3), 0 otherwise.
%F A005075 Additive with a(p^e) = p^2 if p = 2 (mod 3), 0 otherwise.
%F A005076 Additive with a(p^e) = p^3 if p = 2 (mod 3), 0 otherwise.
%F A005077 Additive with a(p^e) = p^4 if p = 2 (mod 3), 0 otherwise.
%F A005078 Additive with a(p^e) = p if p = 1 (mod 4), 0 otherwise.
%F A005079 Additive with a(p^e) = p^2 if p = 1 (mod 4), 0 otherwise.
%F A005080 Additive with a(p^e) = p^3 if p = 1 (mod 4), 0 otherwise.
%F A005081 Additive with a(p^e) = p^4 if p = 1 (mod 4), 0 otherwise.
%F A005082 Additive with a(p^e) = p if p = 3 (mod 4), 0 otherwise.
%F A005083 Additive with a(p^e) = p^2 if p = 3 (mod 4), 0 otherwise.
%F A005084 Additive with a(p^e) = p^3 if p = 3 (mod 4), 0 otherwise.
%F A005085 Additive with a(p^e) = p^4 if p = 3 (mod 4), 0 otherwise.
%F A005087 Additive with a(p^e) = 0 if p = 2, 1 otherwise.
%F A005088 Additive with a(p^e) = 1 if p = 1 (mod 3), 0 otherwise.
%F A005089 Additive with a(p^e) = 1 if p = 1 (mod 4), 0 otherwise.
%F A005090 Additive with a(p^e) = 1 if p = 2 (mod 3), 0 otherwise.
%F A005091 Additive with a(p^e) = 1 if p = 3 (mod 4), 0 otherwise.
%F A005094 Additive with a(p^e) = 0 if p = 2, 1 if p = 1 (mod 4), -1 if 
p = 3 (mod 4).
%F A007814 Totally additive with a(p) = 1 if p = 2, 0 otherwise.
%F A007949 Totally additive with a(p) = 1 if p = 3, 0 otherwise.
%F A008472 Additive with a(p^e) = p.
%F A008474 Additive with a(p^e) = p + e.
%F A008475 Additive with a(p^e) = p^e.
%F A008476 Additive with a(p^e) = e^p.
%F A046660 Additive with a(p^e) = e - 1.
%F A028235 Fraction is additive with a(p^e) = 1/p.
%F A028236 Fraction is additive with a(p^e) = 1/p^e.
%Y A028236 Denominator is n (A000027).
%F A048675 Totally additive with a(p^e) = 2^(PrimePi(p)-1), where 
PrimePi(n) = A000720(n).
%F A056169 Additive with a(p^e) = 1 if e = 1, 0 otherwise.
%F A056170 Additive with a(p^e) = 0 if e = 1, 1 otherwise.
%F A056239 Totally additive with a(p) = PrimePi(p), where PrimePi(n) = 
A000720(n).
%F A059841 Additive with a(p^e) = 1 if p = 2, 0 otherwise.
%F A066328 Additive with a(p^e) = PrimePi(p), where PrimePi(n) = 
A000720(n).
%F A067666 Totally additive with a(p) = p^2.
%F A076649 Totally additive with a(p) = A055642(p).
%F A079978 Additive with a(p^e) = 1 if p = 3, 0 otherwise.
%F A080256 Additive with a(p^e) = e + 1.
%F A081403 Additive with a(p^e) = p^{2e}.
%F A090885 Additive with a(p^e) = e^2.
%F A106490 Additive with a(p^e) = 1 + a(e).
%F A106492 Additive with a(p^e) = p + a(e).
%F A112765 Totally additive with a(p) = 1 if p = 5, 0 otherwise.
%F A113177 Totally additive with a(p) = F(p).
%F A113178 Additive with a(p^e) = F(p).
%F A113222 Additive with a(p^e) = F(p^e).
%F A115357 Sequence shifted right by 2 is additive with a(p^e) = 1 if p 
= 2 or 3, 0 otherwise.
%F A121262 Additive with a(p^e) = 1 if p = 2 and e > 1, 0 otherwise.
%F A121262 Sequence shifted right by 2 is additive with a(p^e) = 1 if p 
= 2 and e = 1, 0 otherwise.

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