|
| |
|
|
A006285
|
|
Odd numbers not of form p + 2^k (de Polignac numbers).
(Formerly M5390)
|
|
39
|
|
|
|
1, 127, 149, 251, 331, 337, 373, 509, 599, 701, 757, 809, 877, 905, 907, 959, 977, 997, 1019, 1087, 1199, 1207, 1211, 1243, 1259, 1271, 1477, 1529, 1541, 1549, 1589, 1597, 1619, 1649, 1657, 1719, 1759, 1777, 1783, 1807, 1829, 1859, 1867, 1927, 1969, 1973, 1985, 2171, 2203, 2213, 2231, 2263, 2279, 2293, 2377, 2429, 2465, 2503, 2579, 2669
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
Crocker shows that this sequence is infinite; in particular, 2^2^n - 5 is in this sequence for each n > 2. - Charles R Greathouse IV, Sep 01 2015
Problem: what is the asymptotic density of de Polignac numbers? Based on the data in A254248, it seems this sequence may have an asymptotic density d > 0.05. Conjecture (cf. Pomerance 2013): the density d(n) of de Polignac numbers <= n is d(n) ~ (1 - 2/log(n))^(log(n)/log(2)), so the asymptotic density d = exp(-2/log(2)) = 0.055833... = 0.111666.../2. - Thomas Ordowski, Jan 30 2021
Romanov (or Romanoff) proved in 1934 that the complementary sequence has a positive lower asymptotic density, and the assumed asymptotic density was later named Romanov's constant (Pintz, 2006).
The lower asymptotic density of this sequence is positive (Van Der Corput, 1950; Erdős, 1950), and larger than 0.00905 (Habsieger and Roblot, 2006).
The upper asymptotic density of this sequence is smaller than 0.392352 (Elsholtz and Schlage-Puchta, 2018).
Previous bounds on the upper asymptotic density were given by Chen and Sun (2006), Pintz (2006), Habsieger and Roblot (2006), Lü (2007) and Habsieger and Sivak-Fischler (2010).
Romani (1983) conjectured that the asymptotic density of this sequence is 0.066... (End)
|
|
|
REFERENCES
|
Clifford A. Pickover, A Passion for Mathematics, John Wiley & Sons, Inc., NJ, 2005, pp. 62 & 300.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. G. Van Der Corput, On de Polignac's conjecture, Simon Stevin, Vol. 27 (1950), pp. 99-105.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, see #127.
|
|
|
LINKS
|
Yong-Gao Chen and Xue-Gong Sun, On Romanoff's constant, Journal of Number Theory, Vol. 106, No. 2 (2004), pp. 275-284.
|
|
|
FORMULA
|
Conjecture: a(n) ~ n*exp(2/log(2)) = n*17.91... - Thomas Ordowski, Feb 02 2021
|
|
|
EXAMPLE
|
127 is in the sequence since 127 - 2^0 = 126, 127 - 2^1 = 125, 127 - 2^2 = 123, 127 - 2^3 = 119, 127 - 2^4 = 111, 127 - 2^5 = 95, and 127 - 2^6 = 63 are all composite. - Michael B. Porter, Aug 29 2016
|
|
|
MAPLE
|
N:= 10000: # to get all terms <= N
P:= select(isprime, {2, seq(i, i=3..N, 2)}):
T:= {seq(2^i, i=0..ilog2(N))}:
R:= {seq(i, i=1..N, 2)} minus {seq(seq(p+t, p=P), t=T)}:
|
|
|
MATHEMATICA
|
Do[ i = 0; l = Ceiling[ N[ Log[ 2, n ] ] ]; While[ ! PrimeQ[ n - 2^i ] && i < l, i++ ]; If[ i == l, Print[ n ] ], {n, 1, 2000, 2} ]
Join[{1}, Select[Range[5, 1999, 2], !MemberQ[PrimeQ[#-2^Range[Floor[ Log[ 2, #]]]], True]&]] (* Harvey P. Dale, Jul 22 2011 *)
|
|
|
PROG
|
(PARI) isA006285(n, i=1)={ bittest(n, 0) && until( isprime(n-i) || n<i<<=1, ); i>n } \\ M. F. Hasler, Jun 19 2008, updated Apr 12 2017
(Haskell)
a006285 n = a006285_list !! (n-1)
a006285_list = filter ((== 0) . a109925) [1, 3 ..]
(Magma) lst:=[]; for n in [1..1973 by 2] do x:=-1; repeat x+:=1; a:=n-2^x; until a lt 1 or IsPrime(a); if a lt 1 then Append(~lst, n); end if; end for; lst; // Arkadiusz Wesolowski, Aug 29 2016
(Python)
from itertools import count, islice
from sympy import isprime
def A006285_gen(startvalue=1): # generator of terms
return filter(lambda n: not any(isprime(n-(1<<i)) for i in range(n.bit_length()-1, -1, -1)), count(max(startvalue+(startvalue&1^1), 1), 2))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,nice,easy
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
More terms from Larry Reeves (larryr(AT)acm.org), Apr 13 2000
|
|
|
STATUS
|
approved
|
| |
|
|
