# puzzling sequence

N. J. A. Sloane njas at research.att.com
Fri Dec 5 13:08:33 CET 2003

```> My question is, why is 24 missing?

Because 2*24 = 3*16 ?

B.

Me:  Brendan, thanks!  I would probably have noticed
that if it had not been 5 in the morning....

So A066724 is OK, A000028 is OK, but the description of A026416 is wrong:

%S A000028 2,3,4,5,7,9,11,13,16,17,19,23,24,25,29,30,31,37,40,41,42,43,47,49,53,54,56,59,61,66,67,70,71,73,78,79,81,83,88,
%T A000028 89,97,101,102,103,104,105,107,109,110,113,114,121,127,128,130,131,135,136,137,138,139,149,151,152,154,157,163,165,167,169,
%U A000028 170,173,174,179,180,181,182,184,186,189,190,191,192,193,195,197,199,211,222,223,227,229,230,231,232,233,238,239,240,241
%N A000028 A 2-way classification of integers: a(1) = 2, a(2) = 3, and for n > 2, a(n) is smallest number > a(n-1) not of form a(i)a(j), i<j<n.

%S A026416 1,2,3,4,5,7,9,11,13,16,17,19,23,24,25,29,30,31,37,40,41,42,43,
%T A026416 47,49,53,54,56,59,61,66,67,70,71,73,78,79,81,83,88,89,97,101,
%U A026416 102,103,104,105,107,109,110,113,114,121,127,128,130,131,135
%N A026416 a(1) = 1, a(2) = 2; and for n > 2, a(n) = least positive integer > a(n-1) and not of the form a(i)*a(j) for 1<=i<j<=n.
(must exclude j=n)

%S A066724 1,2,3,4,5,7,9,11,13,16,17,19,23,25,29,30,31,37,41,43,47,49,53,59,61,
%T A066724 67,71,73,79,81,83,84,89,97,101,103,107,109,113,121,127,128,131,137,
%U A066724 139,149,151,154,157,163,167,169,173,179,180,181,191,193,197,199,211
%N A066724 a(1) = 1; for n > 1, a(n) is the least integer > a(n-1) such that the products a(i)*a(j), for 1 <= i < j <= n, are all distinct.

There is also
%S A026477 1,2,3,4,5,7,9,11,13,16,17,19,23,25,29,31,37,41,43,47,49,53,59,61,67,
%T A026477 71,73,79,81,83,89,97,101,103,107,109,113,120,121,127,131,137,139,149,
%U A026477 151,157,163,167,168,169,173,179,181,191,193,197,199,210,211,216,223
%N A026477 a(n) = least positive integer > a(n-1) and not equal to a(i)*a(j)*a(k) for 1<=i<j<k<n.

NJAS

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