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A006285 Odd numbers not of form p + 2^k (de Polignac numbers).
(Formerly M5390)
36
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

Contains both primes (A065381) and composites (A098237). - Jonathan Vos Post, Jun 19 2008

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

From Amiram Eldar, Feb 03 2021: (Start)

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

Guang-Shi Lü, On Romanoff's constant and its generalized problem, Chinese Advances in Mathematics, Vol. 36, No. 1 (2007), pp. 94-100.

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

T. D. Noe, Table of n, a(n) for n = 1..10000

Yong-Gao Chen and Xue-Gong Sun, On Romanoff's constant, Journal of Number Theory, Vol. 106, No. 2 (2004), pp. 275-284.

Roger Crocker, A theorem concerning prime numbers, Mathematics Magazine, Vol. 34, No. 6 (1961), pp. 316-344.

Yuchen Ding, On a problem of Romanoff type, arXiv:2201.12783 [math.NT], 2022.

Christian Elsholtz and Jan-Christoph Schlage-Puchta, On Romanov's constant, Mathematische Zeitschrift, Vol. 288 (2018), pp. 713-724; alternative link.

Paul Erdős, On integers of the form 2^k + p and some related problems, Summa Brasil. Math., Vol. 2 (1950), p. 113-125.

Laurent Habsieger and Xavier-François Roblot, On integers of the form p+2^k, Acta Arithmetica, Vol. 122, No. 1 (2006), pp. 45-50.

Laurent Habsieger and Jimena Sivak-Fischler, An effective version of the Bombieri-Vinogradov theorem, and applications to Chen's theorem and to sums of primes and powers of two, Archiv der Mathematik, Vol. 95, No. 6 (2010), pp. 557-566.

János Pintz, A note on Romanov's constant, Acta Mathematica Hungarica, Vol. 112, No. 1-2 (2006), pp. 1-14.

Paul Pollack, Not Always Buried Deep: A Second Course in Elementary Number Theory, AMS, 2009, p. 201, exercise 34.

Carl Pomerance, Erdős, van der Corput, and the birth of covering congruences, Joint Mathematics Meetings, Special Session on Covering Congruences, San Diego, CA, January, 2013.

F. Romani, Computations concerning primes and powers of two, Calcolo, Vol. 20 (1983), pp. 319-336.

Nikolai Pavlovich Romanoff, Über einige Sätze der additiven Zahlentheorie, Math. Ann., Vol. 109 (1934), pp. 668-678.

Wikipedia, Romanov's theorem.

FORMULA

A109925(a(n)) = 0. - Reinhard Zumkeller, May 27 2015

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)}:

sort(convert(R, list)); # Robert Israel, Sep 23 2016

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 ..]

-- Reinhard Zumkeller, May 27 2015

(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

CROSSREFS

Cf. A133122, A098237, A065381, A156695, A109925, A118954, A232460, A276417, A254248.

See also A058517, A350957, A350959, A350960.

Sequence in context: A334095 A164966 A178088 * A094933 A156702 A180536

Adjacent sequences:  A006282 A006283 A006284 * A006286 A006287 A006288

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Apr 13 2000

STATUS

approved

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Last modified October 6 12:35 EDT 2022. Contains 357264 sequences. (Running on oeis4.)