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A001359 Lesser of twin primes.
(Formerly M2476 N0982)
496
3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451, 1481, 1487, 1607 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Also, solutions to phi(n + 2) = sigma(n). - Conjectured by Jud McCranie, Jan 03 2001; proved by Reinhard Zumkeller, Dec 05 2002

The set of primes for which the weight as defined in A117078 is 3 gives this sequence except for the initial 3. - Rémi Eismann, 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, Jun 05 2008

It is conjectured that A113910(n+4) = a(n+2) for all n. - Creighton Dement, 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. - Paul Bruckman (pbruckman(AT)hotmail.com), Feb 03 2009

A164292(a(n)) = 1; A010051(a(n)-2) = 0 for n > 1. - Reinhard Zumkeller, Mar 29 2010

From Jonathan Sondow, May 22 2010: (Start)

About 15% of primes < 19000 are the lesser of twin primes. About 26% of Ramanujan primes A104272 < 19000 are the lesser of twin primes.

About 46% of primes < 19000 are Ramanujan primes. About 78% of the lesser of twin primes < 19000 are Ramanujan primes.

A reason for the jumps is in Section 7 of "Ramanujan primes and Bertrand's postulate" and in Section 4 of "Ramanujan Primes: Bounds, Runs, Twins, and Gaps". (End)

Primes generated by sequence A040976. - Odimar Fabeny, Jul 12 2010

Primes of the form 2*n - 3 with 2*n - 1 prime n > 2. Primes of the form (n^2-(n-2)^2)/2 - 1 with (n^2-(n-2)^2)/2 + 1 prime so sum of two consecutive odd numbers/2 - 1. - Pierre CAMI, Jan 02 2012

Solutions of the equation n' + (n+2)' = 2, where n' is the arithmetic derivative of n. - Paolo P. Lava, Dec 18 2012

Conjecture: For any integers n >= m > 0, there are infinitely many integers b > a(n) such that the number sum_{k = m}^n a(k)*b^(n-k) (i.e., (a(m), ..., a(n)) in base b) is prime; moreover, when m = 1 there is such an integer b < (n+6)^2. - Zhi-Wei Sun, Mar 26 2013

Except for the initial 3, all terms are congruent to 5 mod 6. One consequence of this is that no term of this sequence appears in A030459. - Alonso del Arte, May 11 2013

Aside from the first term, all terms have digital root 2, 5, or 8. - J. W. Helkenberg, Jul 24 2013

The sequence provides all solutions to the generalized Winkler conjecture (A051451) aside from all multiples of 6. Specifically, these solutions start from n = 3 as a(n) - 3. This gives 8, 14, 26, 38, 56, ... An example from the conjecture is solution 38 from twin prime pairs (3, 5), (41, 43). - Bill McEachen, May 16 2014

REFERENCES

Milton Abramowitz and Irene 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.

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

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

LINKS

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

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

Chris K. Caldwell, First 100000 Twin Primes

Chris K. Caldwell, Twin Primes

Chris K. Caldwell, Largest known twin primes

Chris K. Caldwell, Twin primes

Chris K. Caldwell, The prime pages

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

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

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

Thomas R. Nicely, Enumeration to 10^14 of the twin primes and Brun's constant, Virginia Journal of Science, 46:3 (Fall, 1995), 195-204.

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

Fred Richman, Generating primes by the sieve of Eratosthenes

P. Shiu, A Diophantine Property Associated with Prime Twins, Experimental mathematics 14 (1) (2005)

Jonathan Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly, 116 (2009) 630-635.

Jonathan Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2

Jonathan Sondow and Emmanuel Tsukerman, The p-adic order of power sums, the Erdos-Moser equation, and Bernoulli numbers, arXiv:1401.0322 [math.NT], 2014; see section 4.

Terence Tao, Obstructions to uniformity and arithmetic patterns in the primes

Eric Weisstein's World of Mathematics, Twin Primes

Index entries for primes, gaps between

FORMULA

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

A001359 = { n | A071538(n-1) = A071538(n)-1 } ; A071538(A001359(n)) = n. - M. F. Hasler, Dec 10 2008

A001359 = { prime(n) : A069830(n) = A087454(n) }. - Juri-Stepan Gerasimov, Aug 23 2011

MAPLE

select(k->isprime(k+2), select(isprime, [$1..1616])); # Peter Luschny, Jul 21 2009

A001359 := proc(n) option remember; if n = 1 then 3; else p := nextprime(procname(n-1)) ; while not isprime(p+2) do p := nextprime(p) ; end do: p ; end if; end proc: # R. J. Mathar, Sep 03 2011

MATHEMATICA

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

a[n_] := a[n] = (p = NextPrime[a[n - 1]]; While[!PrimeQ[p + 2], p = NextPrime[p]]; p); a[1] = 3; Table[a[n], {n, 51}]  (* Jean-François Alcover, Dec 13 2011, after R. J. Mathar *)

nextLesserTwinPrime[p_Integer] := Block[{q = p + 2}, While[NextPrime@ q - q > 2, q = NextPrime@ q]; q]; NestList[nextLesserTwinPrime@# &, 3, 50] (* Robert G. Wilson v, May 20 2014 *)

PROG

From M. F. Hasler, 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)

(MAGMA) [n: n in PrimesUpTo(1610) | IsPrime(n+2)];  // Bruno Berselli, Feb 28 2011

CROSSREFS

Subsequence of A003627.

Cf. A006512 (greater of twin primes), A014574, A001097, A077800, A002822, A040040, A054735, A067829, A082496, A088328, A117078, A117563, A001359, A074822, A071538, A007508, A146214.

Cf. A104272 Ramanujan primes, A178127 Lesser of twin Ramanujan primes, A178128 Lesser of twin primes if it is a Ramanujan prime.

Sequence in context: A093326 * A096292 A181747 A078864 A208574 A023218

Adjacent sequences:  A001356 A001357 A001358 * A001360 A001361 A001362

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane, Apr 30 1991

STATUS

approved

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Last modified October 25 23:29 EDT 2014. Contains 248566 sequences.