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A006285 Odd numbers not of form p + 2^x (de Polignac numbers).
(Formerly M5390)
26
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 (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

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

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

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Clifford A. Pickover, A Passion for Mathematics, John Wiley & Sons, Inc., NJ, 2005, pp. 62 & 300.

D. 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

Roger Crocker, A theorem concerning prime numbers, Mathematics Magazine 34:6 (1961), pp. 316+344.

P. Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 193, ex. 5.1.6, p. 216ff. [alternate (broken?) link]

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)={ n%2 & until( isprime(n-i) | n<i<<=1, ); i>n } \\ - M. F. Hasler, Jun 19 2008

(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.

Sequence in context: A126096 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

Link to Pollack Notes corrected by Rainer Rosenthal, Feb 03 2009

STATUS

approved

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Last modified March 28 22:27 EDT 2017. Contains 284249 sequences.