A249270 -id:A249270 - cp's OEIS frontend

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

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.

A339766 Decimal expansion of Sum_{n>=1} A054541(n)/A076954(n-1).

Original entry on oeis.org

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

Offset: 1

Davide Rotondo, Dec 16 2020

With this constant f(1) and using the formula f(n+1) = (floor(f(n))*(f(n))) - ((floor(f(n)))^2 - floor(f(n))) it is possible to obtain the prime numbers repeated exactly a number of times corresponding to the position of the prime number. That is, 2 once, 3 twice, 5 thrice, etc.

			2.61200074043...

Cf. A054541, A076954, A005145, A249270.

imax:=87;First[RealDigits[N[2+Sum[(Prime[i]-Prime[i-1])/Product[Prime[j-1]^(j-1),{j,2,i}],{i,2,imax}],imax]]] (* Stefano Spezia, Dec 16 2020 *)

Equals 2 + (3-2)/(2) + (5-3)/(2*3^2) + (7-5)/(2*3^2*5^3) + (11-7)/(2*3^2*5^3*7^4) + ...

A368497 Decimal expansion of the fixed point c = S(c) of S(x) = Sum_{k>=1} (prime(k) - x) / Product_{i=1..k-1} prime(i).

Original entry on oeis.org

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

Offset: 1

Antonio Graciá Llorente, Dec 27 2023

S(x) = (1-x)*(1+A064648) + A249270 is linear so the fixed point is unique.

With this constant as h(1) = c, sequence h(n+1) = ceiling(h(n)) * (h(n) - ceiling(h(n)) + c) is real numbers with the property that ceiling(h(n)) = prime(n).

			1.709755124475931301268259070090809...

Cf. A064648, A249270, A000040.

Cf. A341930 (S(3/2)), A340469 (S(2)).

solve(x=1,2,suminf(k=1,(prime(k)-x)/prod(i=1,k-1,prime(i)))-x) \\ Michal Paulovic, Dec 28 2023

Equals (A249270 + A064648 + 1)/(A064648 + 2).

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

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A339766 Decimal expansion of Sum_{n>=1} A054541(n)/A076954(n-1).

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A368497 Decimal expansion of the fixed point c = S(c) of S(x) = Sum_{k>=1} (prime(k) - x) / Product_{i=1..k-1} prime(i).

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