Multiplicative sequences

[David W. Wilson]

As I had threatened in an earlier note, I downloaded the OEIS and ran a check
for potential multiplicative sequences.  For each sufficiently long sequence
(10 elements or more), I checked whether the sequence or the sequence minus its
first element was empirically multiplicative.  By these criteria, I found 706
empirically multiplicative sequences in the OEIS.  Most of these sequences
contain some overt or covert hint that they are multiplicative.

In "note1" attached below, I provide multiplicative formulae for 63 of the
sequences I found.  On request, I can provide a complete list of the 706
sequences which are empirically multiplicative.  Ultimately, I would like to
determine which of these sequences are actually multiplicative and provide the
"mult" keyword where needed.  If possible, I would also like to see a formula
a la "note1" for each multiplicative sequence, if possible.  I don't think I
can finish this project myself, so maybe seqfan can pitch in if they feel so
inclined.

- Dave Wilson
-------------- next part --------------
%F A000005 Multiplicative with a(p^e) = e+1
%F A000010 Multiplicative with a(p^e) = (p-1)*p^(e-1)
%F A000012 Multiplicative with a(p^e) = 1
%F A000026 Multiplicative with a(p^e) = p*e
%F A000027 Multiplicative with a(p^e) = p^e
%F A000034 Multiplicative with a(p^e) = 2 if p even; 1 if p odd
%F A000035 Multiplicative with a(p^e) = p%2
%F A000056 Multiplicative with a(p^e) = (p^2-1)*p^(3e-2)
%F A000082 Multiplicative with a(p^e) = p^(2*e-1)*(p+1);
%F A000086 Multiplicative with a(p^e) = 1 if p = 3 and e = 1; 0 if p = 3 and e > 1; 2 if p == 1 (mod 3); 0 if p == 2 (mod 3)
%F A000089 Multiplicative with a(p^e) = 1 if p = 2 and e = 1; 0 if p = 2 and e > 1; 2 if p == 1 (mod 4); 0 if p == 3 (mod 4)
%F A000188 Multiplicative with a(p^e) = p^[e/2]
%F A000189 Multiplicative with a(p^e) = p^[2e/3]
%F A000190 Multiplicative with a(p^e) = p^[3e/4]
%F A000203 Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1)
%F A000224 Multiplicative with a(p^e) = [p^e/6]+2 if p = 2; [p^(e+1)/(2p+2)]+1 if p > 2
%F A000252 Multiplicative with a(p^e) = (p-1)^2*(p+1)*p^(4e-3)
%F A000265 Multiplicative with a(p^e) = 1 if p = 2, p^e if p > 2
%F A000290 Multiplicative with a(p^e) = p^(2e)
%F A000578 Multiplicative with a(p^e) = p^(3e)
%F A000583 Multiplicative with a(p^e) = p^(4e)
%F A000584 Multiplicative with a(p^e) = p^(5e)
%F A000593 Multiplicative with a(p^e) = 1 if p = 2, (p^(e+1)-1)/(p-1) if p > 2
%F A000688 Multiplicative with a(p^e) = A000041(e)
%F A001014 Multiplicative with a(p^e) = p^(6e)
%F A001015 Multiplicative with a(p^e) = p^(7e)
%F A001016 Multiplicative with a(p^e) = p^(8e)
%F A001017 Multiplicative with a(p^e) = p^(9e)
%F A001157 Multiplicative with a(p^e) = (p^(2e+2)-1)/(p^2-1)
%F A001158 Multiplicative with a(p^e) = (p^(3e+3)-1)/(p^3-1)
%F A001159 Multiplicative with a(p^e) = (p^(4e+4)-1)/(p^4-1)
%F A001160 Multiplicative with a(p^e) = (p^(5e+5)-1)/(p^5-1)
%F A001227 Multiplicative with a(p^e) = 1 if p = 2; e+1 if p > 2
%F A001477 Multiplicative with a(p^e) = p^e
%F A001511 Multiplicative with a(p^e) = e+1 if p = 2; 1 if p > 2
%F A001615 Multiplicative with a(p^e) = (p+1)*p^(e-1)
%F A002131 Multiplicative with a(p^e) = p^e if p = 2; (p^(e+1)-1)/(p-1) if p > 2
%F A002472 Multiplicative with a(p^e) = p^(e-1) if p = 2; (p-2)*p^(e-1) if p > 2
%F A002618 Multiplicative with a(p^e) = (p-1)*p^(2e-1)
%F A003484 Multiplicative with a(p^e) = 2e + a_(e mod 4) if p = 2; 1 if p > 2; where a = (1, 0, 0, 2)
%F A003557 Multiplicative with a(p^e) = p^(e-1)
%F A003958 Multiplicative with a(p^e) = (p-1)^e
%F A003959 Multiplicative with a(p^e) = (p+1)^e
%F A003960 Multiplicative with a(p^e) = [(p+1)/2]^e
%F A003961 Multiplicative with a(p^e) = A000040(A000720(p)+1)^e
%F A003962 Multiplicative with a(p^e) = ((A000040(A000720(p)+1)+1)/2)^e
%F A003963 Multiplicative with a(p^e) = A000720(p)^e
%F A003964 Multiplicative with a(p^e) = A000041(A000720(p)+1)^e
%F A003965 Multiplicative with a(p^e) = A000045(A000720(p)+2)^e
%F A005361 Multiplicative with a(p^e) = e
%F A006519 Multiplicative with a(p^e) = p^e if p = 2; 1 if p > 2
%F A007434 Multiplicative with a(p^e) = p^(2e)-p^(2e-2)
%F A007877 Multiplicative with a(p^e) = 2 if p = 2 and e = 0; 0 if p = 2 and e > 0; 1 if p > 2
%F A007913 Multiplicative with a(p^e) = p^(e mod 2)
%F A007947 Multiplicative with a(p^e) = p
%F A007948 Multiplicative with a(p^e) = p^(min(e, 2))
%F A008454 Multiplicative with a(p^e) = p^(10e)
%F A008455 Multiplicative with a(p^e) = p^(11e)
%F A008456 Multiplicative with a(p^e) = p^(12e)
%F A008473 Multiplicative with a(p^e) = p+e
%F A008477 Multiplicative with a(p^e) = e^p
%F A008683 Multiplicative with a(p^e) = -1 if e = 1; 0 if e > 1
%F A008833 Multiplicative with a(p^e) = p^(2[e/2])