A123396: prime case

[franktaw at netscape.net]

Why start with a(0) = 2?  Better, I think, to start with 0, just as
A123396 does.  Then a(0) = a(1) = 0, a(2) = 2, etc.

Really, with this definition, it is not necessary to mention 0 as a
special case at all:
For n >= 0, a(n) = number of earlier terms each of which, when
added to n, give an X.

When there aren't any earlier terms, the number of earlier terms
with any given property is zero.

General rule: don't introduce initial conditions (or other constraints)
if you don't need to.  If you do need initial conditions, keep them
as simple as possible.

If the idea is important enough, you can introduce variations with
other initial conditions later.  Most cases that qualify for that are
already in the database.

Associated rule: understand the rules for operations on the empty
set - they are not arbitrary.  The general rule is that an operation
on the empty set is the identity element for that operation.  This
maintains the crucial property that F(S Union {x}) = F(F(S),x).

Thus, the sum of the elements in the empty set is zero, their
product is one, their least common multiple is one, and their
greatest common divisor is zero.  Max and min are generally
undefined, so if such a case comes up in your sequence, you
should specify the value.  (If you are dealing only with non-
negative integers, the max of the empty set is zero; but you
should make that explicit.)

Franklin T. Adams-Watters


-----Original Message-----
From: zakseidov at yahoo.com

Dear SeqFans,

Inspired by:
A123396 a(0)=0. a(n) = number of earlier terms each of
which, when added to n, give a triangular number

A1 (not submitted yet, pending gurus' Op):

a(0)=2, at n>0 a(n) = number of earlier terms each of
which, when added to n, give a prime number.

See attached graph
(sorry to bother people not interested in primes).

Qs to gurus:
Is a(n)=0 at some large n?
Is it OK for OEIS?
Thanks, Zak


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