OFFSET
1,1
COMMENTS
The integer parts of the sequence having this constant as starting value and thereafter a(n+1) = (frac(a(n))+1) * floor(a(n)), where floor and frac are integer and fractional part, are exactly the sequence of the composite numbers: see the Grime-Haran Numberphile video for details.
LINKS
James Grime and Brady Haran, 2.920050977316, Numberphile video, Nov 26 2020.
FORMULA
Sum_{k >= 1} (c(k) - 1)/(c(1) * c(2) * ... * c(k-1)), where c(k) is the k-th composite number.
EXAMPLE
4.5892461266379861713581024207350707369274148338616748...
PROG
(Python)
from mpmath import * #high precision computations
#nsum function
from sympy import * # to generate prime numbers
mp.dps = 10000
#function that generates constant that encodes all composite numbers
#cnt - number of prime numbers
def composconst(cnt):
if cnt==1:
return 4-1
primlist=list()
i=0
while (i<cnt):
primlist.append(prime(i+1))
i=i+1
prims=set(primlist)
alllist=range(2, primlist[-1]+2) #all numbers [2..prime(cnt)+1]
alls=set(alllist)
comps=alls-prims #all composite numbers [4..prime(cnt)+1]
complist=list(comps)
cnt2 = len(complist)
return nsum(lambda k: (complist[int(k)]-1)/nprod(lambda l: complist[int(l)], [0, k-1]), [0, cnt2-1])
compconst(50)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Kamil Zabkiewicz, Dec 07 2020
STATUS
approved