OFFSET
1,2
COMMENTS
Conjecturally, a(n) is the smallest number such that n primes occur infinitely often among (x+a(1), ...,x+a(n)).
From M. F. Hasler, Nov 25 2024: (Start)
For a given prime p, if r is the only residue (mod p) not among {a(1), ..., a(n)} (mod p) for some index n, then no term of the sequence can be congruent to r (mod p).
(Instead of a(1...n), one can consider any collection of terms.) - Examples:
(1) p = 2, r = 0, n = 1: No term can be congruent to 0 (mod 2), i.e., even.
(2) p = 3, r = 2, n = 2: No term may be congruent to 2 (mod 3).
(3) p = 5, r = 0, n = 4: No term may be a multiple of 5.
(4) p = 7, r = 4, n = 6: No term may be congruent to 4 (mod 7).
(5) p = 11, r = 6, n = 11: No term may be congruent to 6 (mod 11). (End)
REFERENCES
R. K. Guy's "Unsolved Problems in Number Theory" (2nd edition, Springer, 1994), Section A9.
LINKS
David W. Wilson, Table of n, a(n) for n = 1..10000
Sean A. Irvine, Java program (github)
EXAMPLE
From M. F. Hasler, Nov 25 2024: (Start)
a(2) can't be 2 because otherwise for the prime p = 2, we would have {a(1), a(2)} == {0, 1} (mod p), a complete set of residues. (For the same reason, no other term can be even.) So a(2) = 3 is the smallest possible choice.
Similarly, a(3) must be odd but not congruent to 2 (mod 3) (*), otherwise {a(1), a(2), a(3)} would form a complete set of residues (mod 3). (* As before, this holds for all terms of the sequence.)
So 5 is excluded and the smallest choice is a(3) = 7. (End)
PROG
(PARI) upto(N, a=[1])={for(n=2, N, forstep(k=a[n-1]+2, oo, 2, forprime(p=3, n, #Set(concat(a, k)%p)==p && next(2)); a=concat(a, k); break)); a} \\ M. F. Hasler, Nov 25 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from David Wasserman, Aug 17 2005
Old name has been interchanged with Wasserman's comment, as old name only a conjectural definition of the sequence. Edited by Christopher J. Smyth, May 12 2016
Definition reworded by M. F. Hasler, Nov 25 2024
STATUS
approved