A340469 -id:A340469 - cp's OEIS frontend

A341930 Decimal expansion of (A249270 + A340469)/2.

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

Offset: 1

Davide Rotondo, Feb 23 2021

With this constant r(1) and using the formula r(n+1) = round(r(n))*(r(n) - round(r(n)) + 1.5) it is possible to obtain the sequence of prime numbers because round(r(n)) = prime(n).

			2.06743589142023422951884707566789...

Cf. A249270, A340469.

suminf(k=1, (prime(k)-1.5)/prod(i=1, k-1, prime(i))) \\ Michel Marcus, Feb 23 2021

r(1) = Sum_{k>=1} (prime(k)-1.5)/Product_{i=1..k-1} prime(i).

A064648 Decimal expansion of sum of reciprocals of primorial numbers: 1/2 + 1/6 + 1/30 + 1/210 + ...

Original entry on oeis.org

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

Offset: 0

Labos Elemer, Oct 04 2001

The Engel expansion of this constant is the sequence of primes. - Jonathan Vos Post, May 04 2005
Let S be the operator over the space omega of infinite sequences of numbers, defined to be the Engel expansion of the sum of reciprocals of primorials of a sequence p of numbers; than the eigenvalue-equation S p = p is satisfied by the sequence of prime numbers. - Ralf Steiner, Dec 31 2016
This constant is irrational (Griffiths, 2015). - Amiram Eldar, Oct 27 2020

			0.705230171791800965147431682888248513743577639109154328192267913813919...

Friedrich Engel, "Entwicklung der Zahlen nach Stammbruechen" Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg. pp. 190-191, 1913.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Friedrich Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
Martin Griffiths, 99.29 On the sum of the reciprocals of the primorials, The Mathematical Gazette, Vol. 99, No. 546 (2015), pp. 522-523.
Simon Plouffe, Sum of the product of prime reciprocals.
Eric Weisstein's World of Mathematics, Engel Expansion.

Cf. A002110, A054543, A000027, A053977, A006784, A028259, A165509 (continued fraction).

Cf. A249270, A340469.

RealDigits[ Sum[1/Product[ Prime[i], {i, n}], {n, 58}], 10, 111][[1]] (* Robert G. Wilson v, Aug 05 2005 *)
RealDigits[Total[1/#&/@FoldList[Times,Prime[Range[100]]]],10,120][[1]] (* Harvey P. Dale, Aug 27 2019 *)

default(realprecision, 20080); p=1; s=x=0; for (k=1, 10^9, p*=prime(k); s+=1.0/p; if (s==x, break); x=s ); x*=10; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b064648.txt", n, " ", d)) \\ Harry J. Smith, Sep 21 2009

@CachedFunction
def pv(n):
    a = 1
    b = 0
    for i in (1..n):
        a *= nth_prime(i)
        b += 1/a
    return b
N(pv(100),digits=108) # From Maple code Jani Melik, Jul 22 2015

(1/2)*(1 + (1/3)*(1 + (1/5)*(1 + (1/7)*(1 + (1/11)*(1 + (1/13)*(1 + ...)))))). - Jonathan Sondow, Aug 04 2014

Equals Sum_{n>=1} 1/A002110(n). - Amiram Eldar, Oct 27 2020

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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A341930 Decimal expansion of (A249270 + A340469)/2.

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A064648 Decimal expansion of sum of reciprocals of primorial numbers: 1/2 + 1/6 + 1/30 + 1/210 + ...

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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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