A237261 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.

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, 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, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 5, 7, 6

Offset

1,4

Comments

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

The third zero is a(1622629) = 0.

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.

Links

Lei Zhou, Table of n, a(n) for n = 1..10000

Example

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

Maple

with(numtheory):
a:= n-> add(`if`((f-> not 2 in f and 0<nops(f) and nops(f)<3)
        (factorset(2*n-1-2^k)), 1, 0), k=1..ilog2(2*n-1)):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 06 2014

Mathematica

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}]

Crossrefs

Cf. A005408, A000041.

Sequence in context: A237720 A075172 A347643 * A004258 A029837 A070939

Adjacent sequences: A237258 A237259 A237260 * A237262 A237263 A237264

Keyword

nonn,easy

Author

Lei Zhou, Feb 05 2014

Status

approved