A074171 a(1) = 1. For n >= 2, a(n) is either a(n-1)+n or a(n-1)-n; we use the minus sign only if a(n-1) is prime. E.g., since a(2)=3 is prime, a(3)=a(2)-3=0.

1, 3, 0, 4, 9, 15, 22, 30, 39, 49, 60, 72, 85, 99, 114, 130, 147, 165, 184, 204, 225, 247, 270, 294, 319, 345, 372, 400, 429, 459, 490, 522, 555, 589, 624, 660, 697, 735, 774, 814, 855, 897, 940, 984, 1029, 1075, 1122, 1170, 1219, 1269, 1320, 1372, 1425, 1479

Offset: 1

Author

Amarnath Murthy, Aug 30 2002

Keywords

• easy
• nonn

Comments

In spite of the definition, this is simply 1, 3, then numbers of the form n*(n+7)/2 (A055999). In other words, a(n) = (n-3)*(n+4)/2 for n >= 3. The proof is by induction: For n > 3, a(n-1) = (n-4)*(n+3)/2 is composite, so a(n) = a(n-1) + n = (n-3)*(n+4)/2. - Dean Hickerson, T. D. Noe, Paul C. Leopardi, Labos Elemer and others, Oct 04 2004

If a 2-set Y and a 3-set Z, having one element in common, are subsets of an n-set X then a(n) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Oct 03 2007

Examples

			a(1) = 1
a(2) = a(1) + 2 = 3, which is prime, so
a(3) = a(2) - 3 = 0, which is not prime, so
a(4) = a(3) + 4 = 4, which is not prime, etc.
		

Links

• Milan Janjic, Two Enumerative Functions.

Crossrefs

Cf. A074170, A055999.

Programs

Mathematica

{ta={1, 3}, tb={{0}}};Do[s=Last[ta]; If[PrimeQ[s], ta=Append[ta, s-n]]; If[ !PrimeQ[s], ta=Append[ta, s+n]]; Print[{a=Last[ta], b=(n-3)*(n+4)/2, a-b}]; tb=Append[tb, a-b], {n, 3, 100000}]; {ta, {tb, Union[tb]}} (* Labos Elemer, Oct 07 2004 *)

Formula

a(1) = 1, a(2) = 3; a(n+1) = a(n)+n if a(n) is not a prime; a(n+1) = a(n)-n if a(n) is prime.

Extensions

More terms from Jason Earls, Sep 01 2002

More terms from Labos Elemer, Oct 07 2004