A237261 - OEIS

%I #14 Feb 06 2014 18:35:00

%S 0,0,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,

%T 6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,5,5,6,5,6,6,6,5,6,6,6,6,7,7,7,

%U 6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,6,5,7,6

%N Number of ways to write the n-th odd number in the form of 2^k+p^m*q^h, where p < q are 1 or odd prime numbers, and k, m, h >= 1.

%C Considering 1^m = 1, when p = 1, we always use m = 1.

%C The third zero is a(1622629) = 0.

%C The distribution of the items of this sequence is a bell curve with the mode of 10, which does not likely to grow with n.

%H Lei Zhou, <a href="/A237261/b237261.txt">Table of n, a(n) for n = 1..10000</a>

%e 3rd odd number is A005408(3) = 5 = 2^1+1^1*3^1, 1 case; so a(3) = 1.

%e 10th odd number is A005408(10) = 19 = 2^1+1^1*17^1 = 2^2+3^1*5^1 = 2^3+1^1*7^1 = 2^4+1^1*3^1, 4 cases; so a(10) = 4.

%p with(numtheory):

%p a:= n-> add(`if`((f-> not 2 in f and 0<nops(f) and nops(f)<3)

%p         (factorset(2*n-1-2^k)), 1, 0), k=1..ilog2(2*n-1)):

%p seq(a(n), n=1..100);  # _Alois P. Heinz_, Feb 06 2014

%t Table[ct = 0; m = 1; While[m = m*2; m < n, If[fi = FactorInteger[n - m]; (fi[[1, 1]] >= 3) && (Length[fi] <= 2), ct++]]; ct, {n, 5, 177, 2}]

%Y Cf. A005408, A000041.

%K nonn,easy

%O 1,4

%A _Lei Zhou_, Feb 05 2014