A129699 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 exists.

2, 1, 0, 4, 2, 2, 4, 4, 2, 4, 3, 6, 73, 4, 3, 16, 9, 6, 7, 2, 17, 10, 3, 2, 10, 2, 36, 58, 9, 2, 7, 4, 6, 82, 3, 2, 25, 4, 11, 10, 2, 6, 43, 2, 14, 46, 11, 38, 37, 2, 32, 130, 14, 2, 28, 2, 5, 28, 4, 14, 37, 16, 24, 16, 2, 2, 40, 4, 2, 10, 8, 6, 46, 22, 3, 28, 5, 18, 16, 2, 26, 10, 19, 12, 10, 8

Offset: 0

Author

Jonathan Vos Post, Jun 01 2007

Keywords

• easy
• nonn

Comments

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 0th 4-gonal number is 0th square = 0 and 0 + c^2 cannot be prime for any integer c. B[5,6] = 4 because 6th + 5th pentagonal numbers = 51 + 22 = 73 is prime. B[8,2] = 3 because 3rd + 2nd octagonal numbers = 21 + 8 = 29 is prime.

The sequence of associated primes starts 3, 2, 5, 43, 41, 73, 157, 227, 271, 433, 541, 857, 35107, 1193, 1427,... - R. J. Mathar, Jun 12 2008

Links

• Eric Weisstein's World of Mathematics, Polygonal Number.

Crossrefs

Cf. A000040, A000217, A060354.

Programs

Maple

P := proc(k,n) n/2*((k-2)*n-k+4) ; end: A129699 := proc(n) for m from 0 to 100000 do if isprime(P(n+3,n)+P(n+3,m)) then RETURN(m) ; fi ; od: RETURN(-1) ; end: for n from 0 to 200 do printf("%d,",A129699(n)) ; od: # R. J. Mathar, Jun 12 2008

Formula

a(n) = min{m: m-th (n+3)-gonal number + n-th (n+3)-gonal number is prime}.

Extensions

Corrected and extended by R. J. Mathar, Jun 12 2008