알라디-그린스테드 상수

알라디-그린스테드 상수(Alladi–Grinstead constant)는 알라디 (K. Alladi)와 그린스테드(C. Grinstead)로부터 명명되었다. 알라디-그린스테드 상수는 자연로그의 밑 e에 지수로 작용하는 뤼로스 상수의 1의 보수와 관계있다. n ! {\displaystyle n!} 의 몇몇 초기 팩토리얼을 고려해본다. 가장 작은 소수로부터 자연수의 순서로 대상 정수 n {\displaystyle n} 의 개수대로 재정렬한다. 4 ! = 4 ⋅ 3 ! {\displaystyle 4!=4\cdot 3!} = 4 ⋅ 3 ⋅ 2 {\displaystyle \;\;\;=4\cdot 3\cdot 2} = 2 2 ⋅ 3 ⋅ 2 {\displaystyle \;\;\;=2^{2}\cdot 3\cdot 2} = 2 ⋅ 2 ⋅ 2 ⋅ 3 {\displaystyle \;\;\;=2\cdot 2\cdot 2\cdot 3} 5 ! = 5 ⋅ 4 ! {\displaystyle 5!=5\cdot 4!} = 5 ⋅ 2 ⋅ 3 ⋅ 2 2 {\displaystyle \;\;\;=5\cdot 2\cdot 3\cdot 2^{2}} = 2 ⋅ 3 ⋅ 2 2 ⋅ 5 {\displaystyle \;\;\;=2\cdot 3\cdot 2^{2}\cdot 5} = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5 {\displaystyle \;\;\;=2\cdot 2\cdot 2\cdot 3\cdot 5} 6 ! = 6 ⋅ 5 ! {\displaystyle 6!=6\cdot 5!} = 2 ⋅ 3 ⋅ 5 ! {\displaystyle \;\;\;=2\cdot 3\cdot 5!} = 2 ⋅ 3 ⋅ 2 ⋅ 3 ⋅ 2 2 ⋅ 5 {\displaystyle \;\;\;=2\cdot 3\cdot 2\cdot 3\cdot 2^{2}\cdot 5} = 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 2 2 ⋅ 5 {\displaystyle \;\;\;=2\cdot 2\cdot 3\cdot 3\cdot 2^{2}\cdot 5} = 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 4 ⋅ 5 {\displaystyle \;\;\;=2\cdot 2\cdot 3\cdot 3\cdot 4\cdot 5} 7 ! = 7 ⋅ 6 ! {\displaystyle 7!=7\cdot 6!} = 7 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 2 2 ⋅ 5 {\displaystyle \;\;\;=7\cdot 2\cdot 2\cdot 3\cdot 3\cdot 2^{2}\cdot 5} = 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 2 2 ⋅ 5 ⋅ 7 {\displaystyle \;\;\;=2\cdot 2\cdot 3\cdot 3\cdot 2^{2}\cdot 5\cdot 7} = 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 4 ⋅ 5 ⋅ 7 {\displaystyle \;\;\;=2\cdot 2\cdot 3\cdot 3\cdot 4\cdot 5\cdot 7} 8 ! = 8 ⋅ 7 ! {\displaystyle 8!=8\cdot 7!} = 2 3 ⋅ 7 ! {\displaystyle \;\;\;=2^{3}\cdot 7!} = 2 3 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 2 2 ⋅ 5 ⋅ 7 {\displaystyle \;\;\;=2^{3}\cdot 2\cdot 2\cdot 3\cdot 3\cdot 2^{2}\cdot 5\cdot 7} = 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 2 2 ⋅ 5 ⋅ 7 ⋅ 2 3 {\displaystyle \;\;\;=2\cdot 2\cdot 3\cdot 3\cdot 2^{2}\cdot 5\cdot 7\cdot 2^{3}} = 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 4 ⋅ 5 ⋅ 7 ⋅ 8 {\displaystyle \;\;\;=2\cdot 2\cdot 3\cdot 3\cdot 4\cdot 5\cdot 7\cdot 8} 9 ! = 9 ⋅ 8 ! {\displaystyle 9!=9\cdot 8!} = 3 2 ⋅ 8 ! {\displaystyle \;\;\;=3^{2}\cdot 8!} = 3 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 2 2 ⋅ 5 ⋅ 7 ⋅ 2 3 {\displaystyle \;\;\;=3^{2}\cdot 2\cdot 2\cdot 3\cdot 3\cdot 2^{2}\cdot 5\cdot 7\cdot 2^{3}} = 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 2 2 ⋅ 5 ⋅ 7 ⋅ 2 3 ⋅ 3 2 {\displaystyle \;\;\;=2\cdot 2\cdot 3\cdot 3\cdot 2^{2}\cdot 5\cdot 7\cdot 2^{3}\cdot 3^{2}} = 3 2 ⋅ 3 ⋅ 3 ⋅ 2 2 ⋅ 2 2 ⋅ 5 ⋅ 7 ⋅ 2 3 {\displaystyle \;\;\;=3^{2}\cdot 3\cdot 3\cdot 2^{2}\cdot 2^{2}\cdot 5\cdot 7\cdot 2^{3}} = 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 4 ⋅ 4 ⋅ 5 ⋅ 7 ⋅ 8 {\displaystyle \;\;\;=3\cdot 3\cdot 3\cdot 3\cdot 4\cdot 4\cdot 5\cdot 7\cdot 8} 10 ! = 10 ⋅ 9 ! {\displaystyle 10!=10\cdot 9!} = 2 ⋅ 5 ⋅ 9 ! {\displaystyle \;\;\;=2\cdot 5\cdot 9!} = 2 ⋅ 5 ⋅ 3 2 ⋅ 3 ⋅ 3 ⋅ 2 2 ⋅ 2 2 ⋅ 5 ⋅ 7 ⋅ 2 3 {\displaystyle \;\;\;=2\cdot 5\cdot 3^{2}\cdot 3\cdot 3\cdot 2^{2}\cdot 2^{2}\cdot 5\cdot 7\cdot 2^{3}} = 3 2 ⋅ 3 ⋅ 3 ⋅ 2 2 ⋅ 2 2 ⋅ 2 ⋅ 5 ⋅ 5 ⋅ 7 ⋅ 2 3 {\displaystyle \;\;\;=3^{2}\cdot 3\cdot 3\cdot 2^{2}\cdot 2^{2}\cdot 2\cdot 5\cdot 5\cdot 7\cdot 2^{3}} = 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 2 2 ⋅ 2 3 ⋅ 5 ⋅ 5 ⋅ 7 ⋅ 2 3 {\displaystyle \;\;\;=3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 2^{2}\cdot 2^{3}\cdot 5\cdot 5\cdot 7\cdot 2^{3}} = 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 2 2 ⋅ 5 ⋅ 5 ⋅ 7 ⋅ 2 3 ⋅ 2 3 {\displaystyle \;\;\;=3\cdot 3\cdot 3\cdot 3\cdot 2^{2}\cdot 5\cdot 5\cdot 7\cdot 2^{3}\cdot 2^{3}} = 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 4 ⋅ 5 ⋅ 5 ⋅ 7 ⋅ 8 ⋅ 8 {\displaystyle \;\;\;=3\cdot 3\cdot 3\cdot 3\cdot 4\cdot 5\cdot 5\cdot 7\cdot 8\cdot 8} 팩토리얼 정보가 소수의 제곱의 정보로 이동된다. α ( n ) = ln ⁡ p ln ⁡ n {\displaystyle \alpha (n)={{\ln p} \over {\ln n}}} p = m ( n ) = m a x ( P b e g i n ) {\displaystyle p=m(n)=max\left(P^{begin}\right)} p {\displaystyle p} 는 n ! {\displaystyle n!} 에서 재정렬후 가장 처음에오는 수의 소수제곱정보의 그 소수값 P {\displaystyle P} 이다. α ( 8 ) = ln ⁡ 2 ln ⁡ 8 = ln ⁡ 2 ln ⁡ 2 3 = ln ⁡ 2 3 ln ⁡ 2 = 1 3 = 0.3333 ⋯ {\displaystyle \alpha (8)={{\ln 2} \over {\ln 8}}={{\ln 2} \over {\ln 2^{3}}}={{\ln 2} \over {3\ln 2}}={1 \over 3}=0.3333\cdots } α ( 9 ) = ln ⁡ 3 ln ⁡ 9 = ln ⁡ 3 ln ⁡ 3 2 = ln ⁡ 3 2 ln ⁡ 3 = 1 2 = 0.5 {\displaystyle \alpha (9)={{\ln 3} \over {\ln 9}}={{\ln 3} \over {\ln 3^{2}}}={{\ln 3} \over {2\ln 3}}={1 \over 2}=0.5} n {\displaystyle n} 이 무한히 커지면서 0.8093940205.... {\displaystyle 0.8093940205....} 에 접근한다. lim n → ∞ α ( n ) = e c − 1 = 0.8093940205.... ( O E I S A 085291 ) {\displaystyle \lim _{n\to \infty }\alpha (n)=e^{c-1}=0.8093940205....(OEISA085291)} c = ∑ k = 2 ∞ 1 k ln ⁡ k k − 1 c {\displaystyle c=\sum _{k=2}^{\infty }{1 \over k}\ln {{k} \over {k-1}}\;\;\;c} 는 뤼로스 상수 c = ∑ n = 1 ∞ ζ ( n + 1 ) − 1 n ζ {\displaystyle c=\sum _{n=1}^{\infty }{{\zeta (n+1)-1} \over {n}}\;\;\;\zeta } 는 리만제타함수 = 0.7885305659... ( O E I S A 085361 ) {\displaystyle \;\;\;=0.7885305659...(OEISA085361)} 골롬-딕맨 상수 피보나치 수렴 그래프 수학 상수 “보관된 사본”. 2018년 2월 22일에 원본 문서에서 보존된 문서. 2018년 2월 22일에 확인함.  “보관된 사본”. 2018년 2월 22일에 원본 문서에서 보존된 문서. 2018년 2월 22일에 확인함.  http://mathworld.wolfram.com/Alladi-GrinsteadConstant.html http://mathworld.wolfram.com/Alladi-GrinsteadConstant.html (OEIS의 수열 A085288) (OEIS의 수열 A085289) (OEIS의 수열 A085290)