OFFSET
1,2
COMMENTS
The set of squarefree numbers {A005117} is the lexicographically earliest set of summands {r_i} such that it is possible to get arbitrarily long stretches of squarefree numbers m + r_i.
The probability that k + A005117(i) is squarefree is related to the number of non-unitary prime divisors of k (A056170): the greater the number of such divisors and the smaller the magnitude of the prime, the greater the probability that k + A005117(i) is squarefree when chosen at random from the set of natural numbers (see the Englezou link). For example, within the set {1 + A005117(i)} approximately 53% of numbers are squarefree. Within the set {97560 + A005117(i)} approximately 91% are squarefree.
LINKS
FORMULA
a(n) + 1 is prime by definition.
EXAMPLE
The set of summands {A005117} = 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, ...
a(1) = 1 since 2 is the first prime.
a(2) = 4 since 5 is prime and 6 is semiprime, and this is the earliest set of two such numbers.
a(3) = 192 since 193 is prime, 194 is semiprime, and 195 is sphenic, and this is the earliest set of three such numbers.
a(4) = 38280 since 38281 is prime, 38282 is semiprime, 38283 is sphenic, and 38285 is squarefree with four prime factors, and this is the earliest set of four such numbers.
Table of first 5 terms and their corresponding series along with the exponents of their prime factorizations:
1 | 2: [1]
4 | 5: [1], 6: [1,1]
192 | 193: [1], 194: [1,1], 195: [1,1,1]
38280 | 38281: [1], 38282: [1,1], 38283: [1,1,1], 38285: [1,1,1,1]
97560 | 97561: [1], 97562: [1,1], 97563: [1,1,1], 97565: [1,1,1,1], 97566: [1,1,1,1,1]
PROG
(PARI) a(n) = my(sqf); (sqf(n) = my(c = 0, x = 1); while(c < n, if(issquarefree(x), c++; if(c == n, return(x))); x++)); forprime(p = 1, oo, my(ok = 1); for(m = 1, n, if(omega(p - 1 + sqf(m)) <> m || bigomega(p - 1+ sqf(m)) <> m, ok = 0; break)); if(ok, return(p - 1)))
CROSSREFS
KEYWORD
nonn,more,hard
AUTHOR
Miles Englezou, Oct 02 2025
EXTENSIONS
a(8) from Daniel Suteu, Oct 22 2025
STATUS
approved