A339204 -id:A339204 - cp's OEIS frontend

A339203 Decimal expansion of the generating constant for the exponents of the Mersenne primes.

2, 9, 3, 0, 0, 9, 4, 4, 4, 7, 2, 6, 8, 7, 9, 5, 7, 3, 6, 6, 7, 7, 9, 5, 2, 1, 8, 6, 9, 9, 0, 4, 3, 5, 7, 8, 5, 0, 5, 7, 6, 0, 1, 1, 6, 7, 1, 7, 9, 9, 9, 6, 4, 4, 3, 2, 3, 5, 0, 4, 4, 8, 1, 8, 2, 6, 8, 7, 4, 4, 4, 1, 7, 8, 3, 5, 9, 9, 4, 1, 0, 7, 8, 3, 2, 5, 8, 7

Offset: 1

A.H.M. Smeets, Nov 27 2020

Inspired by the prime generating constant A249270, but here for the exponents of the Mersenne primes, A000043(n).
The producing function is given by f' = floor(f)*(f-floor(f)+1), starting with this constant, f' denoting the next f, and floor(f) being the next term of the sequence being produced by this constant.
Note that this constant is useless in trying to predict the next Mersenne prime exponent. A new known next Mersenne prime exponent will only enable us to calculate this constant more precisely.

			2.93009444726879573667795218699043578505760116717999...

Dylan Friedman, Juli Garbulsky, Bruno Glecer, James Grime, and Massi Tron Florentin, A Prime-Representing Constant, 2019.

Cf. A000043.

Cf. A249270 (for primes), A339204 (for Fibonacci numbers).

Equals Sum_{n > 0} (A000043(n)-1)/(Product_{k = 1..n-1} A000043(k)).

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

Original entry on oeis.org

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

Kamil Zabkiewicz, Dec 07 2020

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.

			4.5892461266379861713581024207350707369274148338616748...

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

Cf. A002808, A249270, A339204.

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

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

cp's OEIS Frontend

A339203 Decimal expansion of the generating constant for the exponents of the Mersenne primes.

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