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A339499
Decimal expansion of the generating constant for the composite numbers.
0
4, 5, 8, 9, 2, 4, 6, 1, 2, 6, 6, 3, 7, 9, 8, 6, 1, 7, 1, 3, 5, 8, 1, 0, 2, 4, 2, 0, 7, 3, 5, 0, 7, 0, 7, 3, 6, 9, 2, 7, 4, 1, 4, 8, 3, 3, 8, 6, 1, 6, 7, 4, 8, 3, 0, 6, 5, 0, 1, 9, 9, 9, 5, 7, 4, 4, 4, 9, 7, 6, 6, 4, 4, 8, 6, 2, 2, 8, 2, 4, 0, 9, 9, 8, 0, 6, 1, 3, 1, 6, 1, 4, 4, 9, 5, 3, 5, 6, 0, 8
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