Proposed Angelini-type sequence, the semiprime analogue of A1221053
Jonathan Post
jvospost3 at gmail.com
Thu Jun 26 00:11:17 CEST 2008
Proposed Angelini-type sequence, the semiprime analogue of A1221053.
Odds are at some point I may have made a mistake, as this was done by
pen and paper, but my draft is:
A sequence S describing the position of its semiprime terms.
4, 1, 5, 6, 9, 10, 8, 14, 15, 21, 12, 22, 16, 25, 26, 33, 18, 34, 20,
35, 38, 39, 24, 46, 49, 51, 28, 55, 30, 57, 32, 62, 65, 69, 74, 37,
77, 82, 85, 86, 43, 87, 45, 91, 93, 48, 94, 95, 52, 106, 111, ...
read as:
"At position 1, there is a semiprime in S, namely 4";
"At position 4, there is a semiprime in S, namely 6";
"At position 5, there is a semiprime in S, namely 9";
"At position 6, there is a semiprime in S, namely 10";
"At position 8, there is a semiprime in S, namely 14";
"At position 9, there is a semiprime in S, namely 15";
"At position 10, there is a semiprime in S, namely 21";
"At position 12, there is a semiprime in S, namely 22";
"At position 14, there is a semiprime in S, namely 25";
"At position 15, there is a semiprime in S, namely 26"; etcetera.
S is built with this greedy rule: when you are about to write a term
of S, always use the smallest positive integer not yet present in S
and not leading to a contradiction.
Thus one cannot start with 1; this would read: "At position 1, there
is a semiprime number in S" [since, 1 is not a semiprime]. Nor with
2, because we start S with the smallest semiprime, namely 4, pointed
to by a(2) = 1. Nor with 3, as that would self-contradictorily index
itself and is not a semiprime. It is consistent to start with a(1) =
4.
S contains all the semiprimes and they appear in their natural order.
Now, does Dean Hickerson's construction, analogized from A121053, work
for the asymptotic value of S?
And is this an interesting sequence to anyone besides myself and Eric Angelini?
Best,
Jonathan Vos Post
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