Numbers of the form 2^(n-1)*(2^n-1)

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A006516 $\scriptstyle 2^{n-1}\,(2^{n}-1),\,n\,\geq \,0.\,$

{0, 1, 6, 28, 120, 496, 2016, 8128, 32640, 130816, 523776, 2096128, 8386560, 33550336, 134209536, 536854528, 2147450880, 8589869056, 34359607296, 137438691328, 549755289600, 2199022206976, 8796090925056, 35184367894528, ...}

Recurrence for numbers of the form 2^(n-1)*(2^n-1)

$a(0)=0,\,$

$a(n)=4a(n-1)+2^{n-1},\quad n\geq 1.\,$

Generating function for numbers of the form 2^(n-1)*(2^n-1)

$G_{\{2^{n-1}\,(2^{n}-1)\}}(x)\equiv \sum _{n=0}^{\infty }2^{n-1}\,(2^{n}-1)~x^{n}={\frac {x}{(1-2x)(1-4x)}}={\frac {x}{1-6x+8x^{2}}}.\,$

Properties of numbers of the form 2^(n-1)*(2^n-1)

Every number of the form $\scriptstyle 2^{n-1}\,(2^{n}-1),\,n\,\geq \,0,\,$ is a triangular number

$2^{n-1}\cdot (2^{n}-1)={\frac {(2^{n}-1)2^{n}}{2}}=t_{2^{n}-1}.\,$

Every number of the form $\scriptstyle 2^{n-1}\,(2^{n}-1),\,n\,\geq \,0,\,$ is also an hexagonal number

$2^{n-1}\cdot (2^{n}-1)=2^{n-1}\cdot (2\cdot 2^{n-1}-1)=h_{2^{n-1}}.\,$

Numbers of the form 2^(p-1)*(2^p-1), p prime

A060286 $\scriptstyle 2^{p-1}\,(2^{p}-1)\,$ where $\scriptstyle p\,$ is a prime number and $\scriptstyle 2^{p}-1\,$ is called a Mersenne number.

{6, 28, 496, 8128, 2096128, 33550336, 8589869056, 137438691328, 35184367894528, 144115187807420416, 2305843008139952128, 9444732965670570950656, 2417851639228158837784576, 38685626227663735544086528, ...}

Even perfect numbers

An even number is a perfect number if and only if it is of the form

$2^{p-1}\,(2^{p}-1),\,$

where $\scriptstyle p\,$ is a prime number and $\scriptstyle 2^{p}-1\,$ is a Mersenne prime.

The even perfect numbers (A000396)

{6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216, ...}

are a subset of A060286, itself a subset of A006516.