

A232667


Primes p such that the pth odious number is prime; odious primes p such that 2p1 is prime.


3



2, 7, 19, 31, 37, 79, 97, 157, 199, 211, 229, 271, 307, 331, 367, 379, 439, 499, 577, 601, 607, 661, 727, 829, 877, 967, 997, 1009, 1069, 1171, 1279, 1459, 1531, 1609, 1627, 1657, 1759, 1867, 2011, 2029, 2131, 2137, 2311, 2551, 2557, 3037, 3061, 3109, 3169, 3181
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OFFSET

1,1


COMMENTS

From Antti Karttunen, Nov 29 & 30 2013: (Start)
This sequence is the intersection of A005382 and A027697.
Proof:
A000069(n) reduces according to the bit parity of n1 as follows:
A000069(n) = 2n  2 when n1 is odious.
A000069(n) = 2n  1 when n1 is evil.
which means that no prime in this sequence can be evil, as then p1 would be an odious number (true for all odd primes) and A000069(p) would be 2(p1) which obviously cannot be a prime, contradicting the requirement. Thus all primes present must belong to the set of odious primes, A027697.
As each prime p here is thus odious, it means that each p1 is an evil number (A001969), and thus A000069(p) = 2p1. And the stipulation that it also must be prime, is just what is required from the terms of A005382. Thus this sequence contains exactly those primes that occur in both A005382 and A027697.
Equally: this is the intersection of A000069 and A005382, thus prime p occurs here iff A000120(p) is odd and 2p1 is prime also.
Also, apart from the first term (2), all the primes (2*a(n))1 are also odious. This follows because for any odd number k, A000120(2k1) = A000120(k).
(End)


LINKS

Antti Karttunen, Table of n, a(n) for n = 1..225


EXAMPLE

7 is a prime and A000069(7) = 13, a prime also, thus 7 is in this sequence.
19 is a prime and A000069(19) = 37, a prime also, thus 19 is in this sequence.
Alternatively:
7 is a prime, 2*71 = 13 is also prime, and when written in binary, 7 = '111', with an odd number of 1bits. Thus 7 is included in this sequence.
The next time this happens, is for 19, as it is a prime, 2*191 = 37 is also prime, and when written in binary, 19 = '10011', also has on odd number of 1bits.


PROG

(Scheme, with Antti Karttunen's IntSeqlibrary and Aubrey Jaffer's SLIBlibrary)
(require 'factor) ;; Includes predicate prime? from SLIBlibrary.
;; Implementation based on the original definition:
(define A232667 (COMPOSE A000040 (MATCHINGPOS 1 1 (lambda (k) (prime? (A000069 (A000040 k)))))))
;; Alternative implementation based on the other definition:
(define A232667 (MATCHINGPOS 1 1 (lambda (n) (and (odd? (A000120 n)) (prime? n) (prime? ( (* 2 n) 1))))))


CROSSREFS

Cf. A000040, A000069, A001969, A000120, A005382, A027697, A092246, A232637.
Sequence in context: A038952 A144589 A179002 * A034794 A213892 A152608
Adjacent sequences: A232664 A232665 A232666 * A232668 A232669 A232670


KEYWORD

nonn


AUTHOR

JuriStepan Gerasimov, Nov 27 2013


EXTENSIONS

Edited and erroneous terms removed by Antti Karttunen, Nov 2930 2013


STATUS

approved



