

A247665


a(1)=2; thereafter a(n) is the smallest number not yet used which is compatible with the condition that a(n) is relatively prime to the next n terms.


23



2, 3, 4, 5, 7, 9, 8, 11, 13, 17, 19, 23, 15, 29, 14, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 25, 27, 79, 83, 16, 49, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 85, 193, 57, 197, 199, 211, 223
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OFFSET

1,1


COMMENTS

It appears that a(k) is even iff k = 2^i1 (cf. A248379). It also appears that all powers of 2 occur in the sequence. (Amarnath Murthy)
The indices of even terms and their values are [1, 2], [3, 4], [7, 8], [15, 14], [31, 16], [63, 32], [127, 64], [255, 128], [511, 122], ...
Will the numbers 6, 10, 21, 22, ... ever occur? 12, 18, 20, ... are also missing, but if 6 never appears then neither will 12, etc.
A related question: are all terms deficient?  Peter Munn, Jul 20 2017


REFERENCES

Amarnath Murthy, Email to N. J. A. Sloane, Oct 05 2014.


LINKS

Russ Cox, Table of n, a(n) for n = 1..100000
Russ Cox, Go program (can compute 5 million terms in about 5 minutes)
Russ Cox, Table of n, a(n) for n = 1..203299677, composite a(n) only; a(203299678) > 2^32.


EXAMPLE

a(1) = 2 must be rel. prime to a(2), so a(2)=3.
a(2) = 3 must be rel. prime to a(3) and a(4), so we can take them to be 4 and 5.
a(3) = 4 must be rel. prime to a(5), a(6), so we must take them to be 7,9.
a(4) = 5 must be rel. prime to a(7), a(8), so we must take them to be 8,11.
At each step after the first, we must choose two new numbers, and we must make sure that not only are they rel. prime to a(n), they are also rel. prime to all a(i), i>n, that have been already chosen.


PROG

(PARI) m=100; v=vector(m); u=vectorsmall(100*m); for(n=1, m, for(i=2, 10^9, if(!u[i], for(j=(n+1)\2, n1, if(gcd(v[j], i)>1, next(2))); v[n]=i; u[i]=1; break))); v \\ Jens Kruse Andersen, Oct 08 2014
(Haskell)
a247665 n = a247665_list !! (n1)
a247665_list = 2 : 3 : f [3] [4..] where
f (x:xs) zs = ys ++ f (xs ++ ys) (zs \\ ys) where
ys = [v, head [w  w < vs, gcd v w == 1]]
(v:vs) = filter (\u > gcd u x == 1 && all ((== 1) . (gcd u)) xs) zs
 Reinhard Zumkeller, Oct 09 2014
(Sage)
# s is the starting point (2 in A247665).
def gen(s):
sequence = [s]
available = list(range(2, 2*s))
available.pop(available.index(s))
yield s
while True:
available.extend(range(available[1]+1, next_prime(available[1])+1))
for i, e in enumerate(available):
if all(gcd(e, sequence[j])==1 for j in range(len(sequence)//2, 0)):
available.pop(i)
sequence.append(e)
yield(e)
break
g = gen(2)
[next(g) for i in range(40)] # (gets first 40 terms of A247665)
# Nadia Heninger, Oct 28 2014


CROSSREFS

Cf. A248379, A248381, A248388, A248389, A248390, A248391, A249049, A249050, A249058, A249556.
Indices of primes and prime powers: A248387, A248918.
Lengths of runs of primes: A249033.
A090252 = similar to A247665 but start with a(1)=1. A249559 starts with a(1)=3.
A249064 is a different generalization.
A064413 is another similar sequence.
Sequence in context: A245820 A105362 A115928 * A117331 A080612 A039261
Adjacent sequences: A247662 A247663 A247664 * A247666 A247667 A247668


KEYWORD

nonn,look


AUTHOR

N. J. A. Sloane, Oct 06 2014 and Oct 08 2014


EXTENSIONS

More terms from Jens Kruse Andersen, Oct 06 2014
Further terms from Russ Cox, Oct 08 2014


STATUS

approved



