A143345 Lexicographically earliest sequence such that a(n) is coprime to the preceding 4 terms (or n-1 terms if n<5) and does not occur earlier.

1, 2, 3, 5, 7, 11, 4, 9, 13, 17, 19, 8, 15, 23, 29, 31, 14, 25, 27, 37, 41, 16, 35, 33, 43, 47, 26, 49, 45, 53, 59, 22, 61, 21, 65, 67, 32, 71, 51, 55, 73, 28, 79, 39, 83, 85, 38, 77, 69, 89, 97, 10, 91, 57, 101, 103, 20, 107, 63, 109, 113, 34, 95, 81, 121, 127, 46, 119, 75, 131, 137, 44, 133, 87, 115, 139, 52, 149, 93, 125, 151, 56, 143, 111, 145, 157, 62, 161, 99, 163, 167, 40, 169, 123, 173, 179, 50, 181

Offset: 1

Author

M. F. Hasler, Jan 18 2011

Keywords

• nonn

Comments

One possible extension of A084937, A103683 to N=4. Here, a(4)=5 is chosen such that a(n) is coprime to a(k) for 0 < k < n <= 4. Another choice is a(k)=k (k<=4), which yieds the different sequence A180348.

It appears that:

- no multiples of 6 occur in this sequence, so it is not a permutation of the integers.

- a(n)=3 (mod 6) iff n=3, n=8, n=13 or n=14+5k, k>0.

- a(n)=0 (mod 2) iff n= 2+5k, k>=0.

- powers of 2 occur in natural order.

- powers of 3 occur in natural order.

- powers of any prime p occur in natural order.

- powers of any number occur in natural order.

Links

• Zak Seidov, Table of n, a(n) for n = 1..5000

Programs

PARI

print1("1,2,3"); a=[1,2,3,L=5]; unused=[4]; v=vector(#a,i,1); for(n=4,99, print1(","a[#a]); for(i=1,#unused,apply(x->gcd(x,unused[i]),a)==v | next; a=concat(vecextract(a,"^1"),unused[i]);unused=vecextract(unused,Str("^",i));next(2));L++;while(apply(x->gcd(x,L),a) !=v,unused=concat(unused,L++-1););a=concat(vecextract(a,"^1"),L))