|
|
A249270
|
|
Decimal expansion of lim_{n->oo} (1/n)*Sum_{k=1..n} smallest prime not dividing k.
|
|
19
|
|
|
2, 9, 2, 0, 0, 5, 0, 9, 7, 7, 3, 1, 6, 1, 3, 4, 7, 1, 2, 0, 9, 2, 5, 6, 2, 9, 1, 7, 1, 1, 2, 0, 1, 9, 4, 6, 8, 0, 0, 2, 7, 2, 7, 8, 9, 9, 3, 2, 1, 4, 2, 6, 7, 1, 9, 7, 7, 2, 6, 8, 2, 5, 3, 3, 1, 0, 7, 7, 3, 3, 7, 7, 2, 1, 2, 7, 7, 6, 6, 1, 2, 4, 1, 9, 0, 1, 7, 8, 1, 1, 2, 3, 1, 7, 5, 8, 3, 7, 4, 2, 2, 9, 8, 3
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
The old definition was "Decimal expansion of the mean value over all positive integers of the least prime not dividing a given integer."
The integer parts of the sequence having this constant as starting value and thereafter x[n+1] = (frac(x[n])+1)*floor(x[n]), where floor and frac are integer and fractional part, are exactly the sequence of the prime numbers: see the Grime-Haran Numberphile video for details. - M. F. Hasler, Nov 28 2020
|
|
REFERENCES
|
Steven R. Finch, Meissel-Mertens constants: Quadratic residues, Mathematical Constants, Cambridge Univ. Press, 2003, pp. 96—98.
|
|
LINKS
|
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 171.
James Grime and Brady Haran, 2.920050977316, Numberphile video, Nov 26 2020.
|
|
FORMULA
|
Sum_{k >= 1} (p_k - 1)/(p_1 p_2 ... p_{k-1}), where p_k is the k-th prime number.
Equals lim_{n->oo} (1/n) * Sum_{k=1..n} A053669(k).
Equals 2 + Sum_{n>=1} (prime(n+1)-prime(n))/prime(n)# = 2 + Sum_{n>=1} A001223(n)/A002110(n). (End)
prime(n+1) = floor(C*prime(n)# - prime(n)*floor(C*prime(n-1)# - 1)) with prime(1)=2 where C is this constant. - Davide Rotondo, Sep 15 2023
|
|
EXAMPLE
|
2.9200509773161347120925629171120194680027278993214267...
|
|
MATHEMATICA
|
digits = 103; Clear[s]; s[m_] := s[m] = Sum[(Prime[k] - 1)/Product[Prime[j], {j, 1, k - 1}] // N[#, digits + 100]&, {k, 1, m}]; s[10]; s[m = 20]; While[RealDigits[s[m]] != RealDigits[s[m/2]], m = 2*m]; RealDigits[s[m], 10, digits] // First
|
|
PROG
|
(Sage)
def sharp_primorial(n): return sloane.A002110(prime_pi(n));
@CachedFunction
def spv(n):
b = 0
for i in (0..n):
b += 1 / sharp_primorial(i)
return b
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|