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

0,1

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

Table of n, a(n) for n=0..85.

Eric Weisstein's World of Mathematics, Polygonal Number.

Formula

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

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

Crossrefs

Cf. A000040, A000217, A060354.

Sequence in context: A091453 A062173 A004558 * A002349 A096794 A106375

Adjacent sequences: A129696 A129697 A129698 * A129700 A129701 A129702

Keyword

easy,nonn

Author

Jonathan Vos Post, Jun 01 2007

Extensions

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

Status

approved