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A235490
Numbers such that none of their prime factors share common 1-bits in the same bit-position and when added (or "ored" or "xored") together, yield a term of A000225 (a binary "repunit").
4
1, 3, 7, 10, 26, 31, 58, 122, 127, 1018, 2042, 8186, 8191, 32762, 131071, 524287, 2097146, 8388602, 33554426, 1073741818, 2147483647, 2305843009213693951, 618970019642690137449562111, 39614081257132168796771975162, 162259276829213363391578010288127, 166153499473114484112975882535043066
OFFSET
1,2
COMMENTS
a(1) = 1 is included on the grounds that it has no prime factors, thus A001414(1)=0, and 0 is one of the terms of A000225, marking the "repunit of length zero".
After 1, the sequence is a union of A000668 (Mersenne primes) and semiprimes of the form 2*A050415. The terms were constructed from the data given in those two entries.
EXAMPLE
7 is included, because it is a prime, and repunit in base-2: '111'.
10 is included, as 10=2*5, and when we add 2 ('10' in binary) and 5 ('101' in binary), we also get 7 ('111' in binary), without producing any carries.
PROG
(Scheme, with Antti Karttunen's IntSeq-library)
(define A235490 (MATCHING-POS 1 1 (lambda (n) (or (= 1 n) (and (= (A001414 n) (A072593 n)) (pow2? (1+ (A001414 n))))))))
(define (pow2? n) (let loop ((n n) (i 0)) (cond ((zero? n) #f) ((odd? n) (and (= 1 n) i)) (else (loop (/ n 2) (1+ i))))))
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Antti Karttunen, Jan 22 2014
STATUS
approved