

A006254


Numbers k such that 2k1 is prime.


77



2, 3, 4, 6, 7, 9, 10, 12, 15, 16, 19, 21, 22, 24, 27, 30, 31, 34, 36, 37, 40, 42, 45, 49, 51, 52, 54, 55, 57, 64, 66, 69, 70, 75, 76, 79, 82, 84, 87, 90, 91, 96, 97, 99, 100, 106, 112, 114, 115, 117, 120, 121, 126, 129, 132, 135, 136, 139, 141, 142, 147, 154, 156, 157
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OFFSET

1,1


COMMENTS

a(n) is the inverse of 2 modulo prime(n) for n >= 2.  JeanFrançois Alcover, May 02 2017
The following sequences (allowing offset of first term) all appear to have the same parity: A034953, triangular numbers with prime indices; A054269, length of period of continued fraction for sqrt(p), p prime; A082749, difference between the sum of next prime(n) natural numbers and the sum of next n primes; A006254, numbers n such that 2n1 is prime; A067076, 2n+3 is a prime.  Jeremy Gardiner, Sep 10 2004
Solutions of the equation (2*k1)'=1, where k' is the arithmetic derivative of k.  Paolo P. Lava, Nov 15 2012
Positions of prime numbers among odd numbers.  Zak Seidov, Mar 26 2013
Also, the integers remaining after removing every third integer following 2, and, recursively, removing every pth integer following the next remaining entry (where p runs through the primes, beginning with 5).  Pete Klimek, Feb 10 2014
Also, numbers k such that k^2 = m^2 + p, for some integers m and every prime p > 2. Applicable m values are m = k  1 (giving p = 2k  1). Less obvious is: no solution exists if m equals any value in A047845, which is the complement of (A006254  1).  Richard R. Forberg, Apr 26 2014
If you define a different type of multiplication (*) where x (*) y = x * y + (x  1) * (y  1), (which has the commutative property) then this is the set of primes that follows.  Jason Atwood, Jun 16 2019


LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000


FORMULA

a(n) = (A000040(n+1) + 1)/2 = A067076(n1) + 2 = A086801(n1)/2 + 2.
a(n) = (1 + A065091(n))/2.  Omar E. Pol, Nov 10 2007
a(n) = sqrt((A065091^2 + 2*A065091+1)/4).  Eric Desbiaux, Jun 29 2009
a(n) = A111333(n+1).  Jonathan Sondow, Jan 20 2016


MATHEMATICA

Rest@Prime@Range@70/2 + 1/2 (* Robert G. Wilson v, Jun 16 2006 *)
Select[Range[200], PrimeQ[2#1]&] (* Harvey P. Dale, Apr 06 2014 *)
a[n_] := ModularInverse[2, Prime[n+1]]; Table[a[n], {n, 1, 100}] (* JeanFrançois Alcover, May 02 2017 *)


PROG

(MAGMA) [n: n in [0..1000]  IsPrime(2*n1)] // Vincenzo Librandi, Nov 18 2010
(PARI) a(n)=prime(n+1)\2+1 \\ Charles R Greathouse IV, Mar 20 2013


CROSSREFS

Cf. A000040, A067076, A086801.
Equals A005097 + 1. A130291 is an essentially identical sequence.
Cf. A065091.
Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).
Numbers n such that 2nk is prime: this seq(k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).
Sequence in context: A225819 A205805 A246372 * A111333 A047701 A164528
Adjacent sequences: A006251 A006252 A006253 * A006255 A006256 A006257


KEYWORD

nonn,easy


AUTHOR

Marc LeBrun


EXTENSIONS

More terms from Erich Friedman
More terms from Omar E. Pol, Nov 10 2007


STATUS

approved



