A130882 a(n) = smallest k such that A002808(n+1) = A002808(n) + (A002808(n) mod k), or 0 if no such k exists.

0, 4, 7, 2, 4, 5, 13, 2, 7, 4, 19, 2, 4, 23, 2, 5, 2, 13, 4, 31, 2, 3, 2, 17, 37, 2, 19, 4, 43, 2, 4, 47, 2, 7, 2, 5, 53, 2, 5, 2, 4, 29, 61, 2, 3, 2, 4, 67, 2, 4, 5, 73, 2, 3, 2, 4, 79, 2, 4, 83, 2, 5, 2, 43, 89, 2, 7, 2, 3, 2, 47, 97

Offset: 1

Author

Rémi Eismann, Aug 21 2007 - Jan 09 2011

Keywords

• nonn

Comments

a(n) is the "weight" of composite numbers.

The decomposition of composite numbers into weight * level + gap is A002808(n) = a(n) * A179621(n) + A073783(n) if a(n) > 0.

Examples

			For n = 1 we have A002808(n) = 4, A002808(n+1) = 6; there is no k such that 6 - 4 = 2 = (4 mod k), hence a(1) = 0.
For n = 3 we have A002808(n) = 8, A002808(n+1) = 9; 7 is the smallest k such that 9 - 8 = 1 = (8 mod k), hence a(3) = 7.
For n = 24 we have A002808(n) = 36, A002808(n+1) = 38; 17 is the smallest k such that 38 - 36 = 2 = (36 mod k), hence a(24) = 17.
		

Links

• Remi Eismann, Table of n, a(n) for n=1..9999

Crossrefs

Cf. A020639, A117078, A117563, A001223, A118534, A090369, A090368, A130533, A130650, A130703.