A123396: prime case

[Rob Pratt]

Regarding max and min of an empty set of real numbers, the conventions

max {} = -infinity,
min {} = infinity 

preserve your crucial property and specialize correctly for the nonnegative integers.

Rob

-----Original Message-----
From: franktaw at netscape.net [mailto:franktaw at netscape.net] 
Sent: Monday, October 16, 2006 11:11 AM
To: seqfan at ext.jussieu.fr
Subject: Re: A123396: prime case

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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