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A184829 a(n) = smallest k such that A000961(n+1) = A000961(n) + (A000961(n) mod k), or 0 if no such k exists.

%I A184829 #17 Oct 18 2024 10:49:44
%S A184829 2,0,2,3,3,2,7,7,3,5,3,3,5,3,23,5,3,2,9,11,3,13,3,5,47,3,29,61,7,3,67,
%T A184829 7,79,7,9,31,3,9,3,5,15,9,3,2,5,25,3,43,3,29,151,53,3,5,167,3,19,3,7,
%U A184829 3,17,199,73,3,5,227,3,239,47,6,3,251,257,3,53,7,3,277,5
%N A184829 a(n) = smallest k such that A000961(n+1) = A000961(n) + (A000961(n) mod k), or 0 if no such k exists.
%C A184829 a(n) is the "weight" of prime powers.
%C A184829 The decomposition of prime powers into weight*level + gap is A000961(n) = a(n)*A184831(n) + A057820(n) if n > 2 and a(n) > 0. [amended by _Jason Yuen_, Oct 17 2024]
%H A184829 Jason Yuen, <a href="/A184829/b184829.txt">Table of n, a(n) for n = 1..10000</a> (correcting Rémi Eismann's previous b-file)
%e A184829 For n = 1 we have A000961(1) = 1, A000961(2) = 2; 2 is the smallest k such that 2 = 1 + (1 mod k), hence a(1) = 2.
%e A184829 For n = 3 we have A000961(3) = 3, A000961(4) = 4; 2 is the smallest k such that 4 = 3 + (3 mod k), hence a(3) = 2.
%e A184829 For n = 24 we have A000961(24) = 49, A000961(25) = 53; 5 is the smallest k such that 53 = 49 + (49 mod k), hence a(24) = 5.
%Y A184829 Cf. A000961, A057820, A184831, A184830, A117078, A117563, A001223, A118534.
%K A184829 nonn
%O A184829 1,1
%A A184829 _Rémi Eismann_, Jan 23 2011
%E A184829 a(1) corrected by _Jason Yuen_, Oct 17 2024