%I A001359 M2476 N0982
%S A001359 3,5,11,17,29,41,59,71,101,107,137,149,179,191,197,227,239,269,281,311,
%T A001359 347,419,431,461,521,569,599,617,641,659,809,821,827,857,881,1019,1031,
%U A001359 1049,1061,1091,1151,1229,1277,1289,1301,1319,1427,1451,1481,1487,1607
%N A001359 Lesser of twin primes.
%C A001359 Also, solutions to phi(n + 2) = sigma(n). - Conjectured by Jud McCranie
(j.mccranie(AT)comcast.net), Jan 03 2001; proved by Reinhard Zumkeller
(reinhard.zumkeller(AT)gmail.com), Dec 05 2002
%C A001359 Primes for which the weight as defined in A117078 is 3 gives this sequence
except for the initial 3. - Remi Eismann (reismann(AT)free.fr), Feb
15 2007
%C A001359 The set of lesser of twin primes larger than three is a proper subset
of the set of primes of the form 3n - 1 (A003627). - Paul Muljadi
(paulmuljadi(AT)yahoo.com), Jun 05 2008
%C A001359 It is conjectured that A113910(n+4) = a(n+2) for all n. [From Creighton
Dement (creighton.k.dement(AT)uni-oldenburg.de), Jan 15 2009]
%C A001359 I would like to conjecture that if f(x) is a series whose terms are x^n,
where n represents the terms of sequence A001359, and if we inspect
{f(x)}^5, the conjecture is that every term of the expansion, say
a_n * x^n, where n is odd and at least equal to 15, has a_n >= 1
. This is not true for {f(x)}^k, k = 1, 2, 3 or 4, but appears to
be true for k >= 5 . [From Paul Bruckman (pbruckman(AT)hotmail.com),
Feb 03 2009]
%C A001359 Largest prime<nth isolated composite. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru),
Nov 07 2009]
%D A001359 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A001359 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A001359 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions,
National Bureau of Standards Applied Math. Series 55, 1964 (and various
reprintings), p. 870.
%D A001359 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag,
1976, page 6.
%D A001359 Harvey Dubner, Twin Prime Statistics, Journal of Integer Sequences, Vol.
8 (2005), Article 05.4.2.
%D A001359 A. Granville and G. Martin, Prime number races, Amer. Math. Monthly,
113 (No. 1, 2006), 1-33.
%D A001359 T. R. Nicely, Enumeration to 10^14 of the twin primes and Brun's constant,
Virginia Journal of Science, 46:3 (Fall, 1995), 195-204.
%H A001359 C. K. Caldwell, <a href="b001359.txt">Table of n, a(n) for n = 1..100000</
a>
%H A001359 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National
Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972
[alternative scanned copy].
%H A001359 C. K. Caldwell, <a href="http://www.utm.edu/research/primes/lists/small/
100ktwins.txt">First 100000 Twin Primes</a>
%H A001359 C. K. Caldwell, <a href="http://www.utm.edu/research/primes/lists/top20/
twin.html">Twin Primes</a>
%H A001359 C. K. Caldwell, <a href="http://www.utm.edu/research/primes/largest.html#biggest">
Largest known twin primes</a>
%H A001359 C. K. Caldwell, <a href="http://primes.utm.edu/glossary/page.php?sort=TwinPrime">
Twin primes</a>
%H A001359 C. K. Caldwell, <a href="http://www.utm.edu/research/primes/">The prime
pages</a>
%H A001359 A. Granville and G. Martin, <a href="http://www.arXiv.org/abs/math.NT/
0408319">Prime number races</a>
%H A001359 Thomas R. Nicely, <a href="http://www.trnicely.net/">Home page, which
has extensive tables.</a>
%H A001359 O. E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica
de los numeros primos y perfectos</a>.
%H A001359 F. Richman, <a href="http://www.math.fau.edu/Richman/primes.htm">Generating
primes by the sieve of Eratosthenes</a>
%H A001359 P. Shiu, <a href="http://www.expmath.org/expmath/volumes/14/14.1/Shiu.pdf">
A Diophantine Property Associated with Prime Twins</a>
%H A001359 T. Tao, <a href="http://arXiv.org/abs/math.NT/0505402">Obstructions to
uniformity and arithmetic patterns in the primes</a>
%H A001359 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
TwinPrimes.html">Link to a section of The World of Mathematics.</
a>
%H A001359 <a href="Sindx_Pri.html#gaps">Index entries for primes, gaps between</
a>
%F A001359 A001359 = { n | A071538(n-1) = A071538(n)-1 } ; A071538(A001359(n)) =
n. [From M. F. Hasler (MHasler(AT)univ-ag.fr), Dec 10 2008]
%p A001359 for i from 1 to 253 do if ithprime(i+1) = ithprime(i) + 2 then print({ithprime(i)});
fi; od; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 19 2007
%p A001359 select(k->isprime(k+2),select(isprime,[$1..1616])); [From Peter Luschny
(peter(AT)luschny.de), Jul 21 2009]
%t A001359 Select[ Prime[ Range[ 253]], PrimeQ[ # + 2] &] (from Robert G. Wilson
v (rgwv(AT)rgwv.com), Jun 09 2005)
%o A001359 Contribution from M. F. Hasler (MHasler(AT)univ-ag.fr), Dec 10 2008:
(Start)
%o A001359 (PARI) A001359(n,p=3) = { while( p+2 < (p=nextprime( p+1 )) | n-->0,);
p-2}
%o A001359 /* The following gives a reasonably good estimate for any value of n
from 1 to infinity ; compare to A146214. */
%o A001359 A001359est(n) = solve( x=1,5*n^2/log(n+1), 1.320323631693739*intnum(t=2.02,
x+1/x,1/log(t)^2)-log(x) +.5 - n)
%o A001359 /* The constant is A114907; the expression in front of +.5 is an estimate
for A071538(x) */ (End)
%Y A001359 Cf. A006512 (greater of twin primes), A014574, A001097, A077800.
%Y A001359 a(n)=A077800(2n-1).
%Y A001359 Cf. A002822, A040040, A054735, A067829, A082496, A088328.
%Y A001359 Cf. A117078, A117563, A001359, A074822.
%Y A001359 Cf. A003627.
%Y A001359 Cf. A071538, A007508, A146214. [From M. F. Hasler (MHasler(AT)univ-ag.fr),
Dec 10 2008]
%Y A001359 Sequence in context: A063700 A078859 A054799 this_sequence A096292 A078864
A023218
%Y A001359 Adjacent sequences: A001356 A001357 A001358 this_sequence A001360 A001361
A001362
%K A001359 nonn,nice,easy
%O A001359 1,1
%A A001359 N. J. A. Sloane (njas(AT)research.att.com).
%E A001359 Corrected comment and added conjecture [From Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de),
Jan 15 2009]
|
