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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 (list; constant; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..100.

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

Cf. A002808, A249270, A339204.

Sequence in context: A104883 A154885 A292192 * A295150 A042956 A128217

Adjacent sequences:  A339496 A339497 A339498 * A339500 A339501 A339502

KEYWORD

nonn,cons

AUTHOR

Kamil Zabkiewicz, Dec 07 2020

STATUS

approved

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Last modified May 7 10:17 EDT 2021. Contains 343649 sequences. (Running on oeis4.)