A129755 (??!!)

Jonathan Post jvospost3 at gmail.com
Fri Jun 1 20:14:39 CEST 2007

```The further generalization that I mentioned to Zak and seqfans was
just submitted as:

NEW SEQUENCE FROM Jonathan Vos Post

%I A000001
%S A000001 2, 2, 0, 4, 2, 2
%N A000001 Least nonnegative m such that P(n+3,n) + P(n+3,m) is prime
where P(k,n) is n-th k-gonal number, or -1 if no such value.
%C A000001 Define array A[k,n] for k>2, n>=0, where A[k,n] = n-th
k-gonal number =
k((n-2)*k - (n-4))/2. Then define array B[k,n] = least m such that the
A[k,n] + m-th k-gonal number is prime.  This sequence is the main
diagonal of B.
The array B[k,n] begins:
k.|.B[k,n]

3.|..2.1.0.1.1.7.4.1.1.7.3...

4.|.-1.2.1.2.1.2.5.2.3.4.1...

5.|..2.3.0.1.1.3.4.1.4.3...

6.|.-1.1.1.4.1.4.4.2.7.4...

7.|..2.3.0.1.2.3.7.1.1.4...

8.|.-1.4.3.2.1.2.1.4.3.2...

B[4,0] = -1 because 0-th 4-gonal number is 0-th square = 0, and 0 +
c^2 cannot be prime for any integer c.

B[5,6] = 4 because 6-th + 5-th pentagonal numbers = 51 + 22 = 73 is prime.

B[8,2] = 3 because 3rd + 2nd octagonal numbers = 21 + 8 = 29 is prime.
%H A000001 Eric W. Weisstein, <a
href="http://mathworld.wolfram.com/PolygonalNumber.html">Polygonal
Number.</a> From MathWorld--A Wolfram Web Resource.
%F A000001 a(n) = min{m: m-th (n+3)-gonal number + n-th (n+3)-gonal
number is prime}.
%Y A000001 Cf. A000040, A000217, A060354.
%O A000001 0,1
%K A000001 ,easy,more,sign,
%A A000001 Jonathan Vos Post (jvospost2 at yahoo.com), Jun 01 2007

The earlier sequence that I submitted, which Tony Noe and Peter Pein
corrected, and Tony Noe is editing, is the k=3 row of the array above.
The k=4 row is surprisingly not in OEIS (unless I missed it
erroneously): B[4,n] = Min{m: n^2 + m^2 is prime}.

Are there other "-1" values in B other than at B[2*i,0]?

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