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Prime Signature

المؤلف:  Sloane, N. J. A.

المصدر:  Sequences A118914 in "The On-Line Encyclopedia of Integer Sequences."

الجزء والصفحة:  ...

15-9-2020

1411

Prime Signature

The prime signature of a positive integer  is a sorted list of nonzero exponents  in the prime factorization

By definition, the prime signature of 1 is {1}" src="https://mathworld.wolfram.com/images/equations/PrimeSignature/Inline3.gif" style="height:15px; width:17px" />. The prime numbers have prime signature {1}" src="https://mathworld.wolfram.com/images/equations/PrimeSignature/Inline4.gif" style="height:15px; width:17px" /> and squares of prime numbers have prime signature {2}" src="https://mathworld.wolfram.com/images/equations/PrimeSignature/Inline5.gif" style="height:15px; width:17px" />.

The following table gives the prime signatures of the first few positive integers (OEIS A118914).

	factorization	prime signature		factorization	prime signature
1	1	{1}" src="https://mathworld.wolfram.com/images/equations/PrimeSignature/Inline8.gif" style="height:15px; width:17px" />	11	11	{1}" src="https://mathworld.wolfram.com/images/equations/PrimeSignature/Inline9.gif" style="height:15px; width:17px" />
2	2	{1}" src="https://mathworld.wolfram.com/images/equations/PrimeSignature/Inline10.gif" style="height:15px; width:17px" />	12		{1,2}" src="https://mathworld.wolfram.com/images/equations/PrimeSignature/Inline12.gif" style="height:15px; width:32px" />
3	3	{1}" src="https://mathworld.wolfram.com/images/equations/PrimeSignature/Inline13.gif" style="height:15px; width:17px" />	13	13	{1}" src="https://mathworld.wolfram.com/images/equations/PrimeSignature/Inline14.gif" style="height:15px; width:17px" />
4		{2}" src="https://mathworld.wolfram.com/images/equations/PrimeSignature/Inline16.gif" style="height:15px; width:17px" />	14		{1,1}" src="https://mathworld.wolfram.com/images/equations/PrimeSignature/Inline18.gif" style="height:15px; width:32px" />
5	5	{1}" src="https://mathworld.wolfram.com/images/equations/PrimeSignature/Inline19.gif" style="height:15px; width:17px" />	15		{1,1}" src="https://mathworld.wolfram.com/images/equations/PrimeSignature/Inline21.gif" style="height:15px; width:32px" />
6		{1,1}" src="https://mathworld.wolfram.com/images/equations/PrimeSignature/Inline23.gif" style="height:15px; width:32px" />	16		{4}" src="https://mathworld.wolfram.com/images/equations/PrimeSignature/Inline25.gif" style="height:15px; width:17px" />
7	7	{1}" src="https://mathworld.wolfram.com/images/equations/PrimeSignature/Inline26.gif" style="height:15px; width:17px" />	17	17	{1}" src="https://mathworld.wolfram.com/images/equations/PrimeSignature/Inline27.gif" style="height:15px; width:17px" />
8		{3}" src="https://mathworld.wolfram.com/images/equations/PrimeSignature/Inline29.gif" style="height:15px; width:17px" />	18		{1,2}" src="https://mathworld.wolfram.com/images/equations/PrimeSignature/Inline31.gif" style="height:15px; width:32px" />
9		{2}" src="https://mathworld.wolfram.com/images/equations/PrimeSignature/Inline33.gif" style="height:15px; width:17px" />	19	19	{1}" src="https://mathworld.wolfram.com/images/equations/PrimeSignature/Inline34.gif" style="height:15px; width:17px" />
10		{1,1}" src="https://mathworld.wolfram.com/images/equations/PrimeSignature/Inline36.gif" style="height:15px; width:32px" />	20		{1,2}" src="https://mathworld.wolfram.com/images/equations/PrimeSignature/Inline38.gif" style="height:15px; width:32px" />

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REFERENCES:

Sloane, N. J. A. Sequences A118914 in "The On-Line Encyclopedia of Integer Sequences."