|
%I #10 Aug 18 2015 00:35:00
%S 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,
%T 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,
%U 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
%N 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.
%C 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.
%C 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
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PolygonalNumber.html">Polygonal Number.</a>
%F a(n) = min{m: m-th (n+3)-gonal number + n-th (n+3)-gonal number is prime}.
%p 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
%Y Cf. A000040, A000217, A060354.
%K easy,nonn
%O 0,1
%A _Jonathan Vos Post_, Jun 01 2007
%E Corrected and extended by _R. J. Mathar_, Jun 12 2008
|
