1,1

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

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

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

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]

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]

Largest prime<nth isolated composite. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 07 2009]

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

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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.

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 6.

Harvey Dubner, Twin Prime Statistics, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.2.

A. Granville and G. Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.

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.

C. K. Caldwell, Table of n, a(n) for n = 1..100000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

C. K. Caldwell, First 100000 Twin Primes

C. K. Caldwell, Twin Primes

C. K. Caldwell, Largest known twin primes

C. K. Caldwell, Twin primes

C. K. Caldwell, The prime pages

A. Granville and G. Martin, Prime number races

Thomas R. Nicely, Home page, which has extensive tables.

O. E. Pol, Determinacion geometrica de los numeros primos y perfectos.

F. Richman, Generating primes by the sieve of Eratosthenes

P. Shiu, A Diophantine Property Associated with Prime Twins

T. Tao, Obstructions to uniformity and arithmetic patterns in the primes

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Index entries for primes, gaps between

A001359 = { n | A071538(n-1) = A071538(n)-1 } ; A071538(A001359(n)) = n. [From M. F. Hasler (MHasler(AT)univ-ag.fr), Dec 10 2008]

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

select(k->isprime(k+2), select(isprime, [$1..1616])); [From Peter Luschny (peter(AT)luschny.de), Jul 21 2009]

Select[ Prime[ Range[ 253]], PrimeQ[ # + 2] &] (from Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 09 2005)

Contribution from M. F. Hasler (MHasler(AT)univ-ag.fr), Dec 10 2008: (Start)

(PARI) A001359(n, p=3) = { while( p+2 < (p=nextprime( p+1 )) | n-->0, ); p-2}

/* The following gives a reasonably good estimate for any value of n from 1 to infinity ; compare to A146214. */

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)

/* The constant is A114907; the expression in front of +.5 is an estimate for A071538(x) */ (End)

Cf. A006512 (greater of twin primes), A014574, A001097, A077800.

a(n)=A077800(2n-1).

Cf. A002822, A040040, A054735, A067829, A082496, A088328.

Cf. A117078, A117563, A001359, A074822.

Cf. A003627.

Cf. A071538, A007508, A146214. [From M. F. Hasler (MHasler(AT)univ-ag.fr), Dec 10 2008]

Sequence in context: A063700 A078859 A054799 this_sequence A096292 A078864 A023218

Adjacent sequences: A001356 A001357 A001358 this_sequence A001360 A001361 A001362

nonn,nice,easy,new

N. J. A. Sloane (njas(AT)research.att.com).

Corrected comment and added conjecture [From Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Jan 15 2009]