A339499 Decimal expansion of the generating constant for the composite numbers.

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

Author

Kamil Zabkiewicz, Dec 07 2020

Keywords

• nonn
• cons

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.

Examples

			4.5892461266379861713581024207350707369274148338616748...
		

Links

• James Grime and Brady Haran, 2.920050977316, Numberphile video, Nov 26 2020.

Crossrefs

Cf. A002808, A249270, A339204.

Programs

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
				

Formula

Sum_{k >= 1} (c(k) - 1)/(c(1) * c(2) * ... * c(k-1)), where c(k) is the k-th composite number.