OFFSET
1,2
COMMENTS
Numbers A005117(k) - 1 such that the length m of the series {A005117(k) - 1 + A005117(i)}, 1 <= i <= m, is a record.
The corresponding A389412(k) for a(1) to a(10) are: 2, 3, 4, 11, 38, 46, 54, 60, 105, 127.
The series of non-unitary prime divisors for a(1) to (10) is nondecreasing (0, 1, 1, 1, 2, 2, 3, 3, 3, 4). This is to be expected as, for some x, the more non-unitary prime divisors x has and the smaller the size of the dividing primes, the greater the probability of choosing a number s = x + A005117(i) and for s to be squarefree (see the Englezou link). For example, within the set {1455300 + A005117(i)} approximately 97% of numbers are squarefree.
LINKS
EXAMPLE
The second term is 4 since 5 is the 4th squarefree number, and A389412(4) = 3, and 3 is greater than any term prior to A389412(4).
The third term is 12 since 13 is the 9th squarefree number, and A389412(9) = 4, and 4 is greater than any term prior to A389412(9).
The fourth term is 28 since 29 is the 18th squarefree number, and A389412(18) = 11, and 11 is greater than any term prior to A389412(18).
Table of the corresponding series for the first four terms:
1: [2, 3]
4: [5, 6, 7]
12: [13, 14, 15, 17]
28: [29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 43]
PROG
(PARI) a(n) = my(f); (f(n) = my(nb = 0); for(r = n, oo, for(m = 1, oo, if(issquarefree(m), if(issquarefree(r + m - 1), nb++, return(nb)))))); my(T = [f(2)], U = [2]); for(t = 2, oo, if(f(t) > T[#T], T = concat(T, f(t)); U = concat(U, t)); if(#T == n, return(U[#U] - 1)))
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Miles Englezou, Oct 03 2025
EXTENSIONS
a(11)-a(14) from Sean A. Irvine, Oct 16 2025
STATUS
approved