A361085 Least prime p > prime(n) such that at least one of p * prime(n)# +- 1 is not squarefree, where prime(n)# is the n-th primorial A002110(n).

3, 5, 29, 31, 139, 167, 43, 127, 211, 41, 607, 1223, 71, 769, 1549, 947, 269, 1129, 163, 577, 673, 107, 4057, 1979, 433, 3833, 4177, 383, 1723, 409, 2399, 4517, 3803, 3061, 3299, 457, 3779, 971, 5749, 2843, 13709

Offset

0,1

Comments

It appears that a product P of distinct primes rarely has the property that P +- 1 has a square factor, and this is even more rare when P has all of the first n primes as factor. This sequence is one possible way to quantify this observation. (One could also display the gap between a(n) and prime(n), or consider b(n) the least product of distinct primes > prime(n) that yields a product with the desired property.)

See also Zumkeller's contested comment in A007018 and the discussion in the linked MathOverflow page.

Links

Table of n, a(n) for n=0..40.

Dan Asimov, Interesting sequence on MathOverflow, math-fun mailing list, Mar 28 2023.

Fredrick M. Nelson, Does a(0)=6, a(n+1)=a(n)^3-a(n), define a square-free sequence?, MathOverflow, Mar 24 2023.

Example

a(0) = 3 because for P = (product of the first 0 primes) = 1, p = 3 is the least prime such that p*P + 1 = 4 = 2^2 is a square; for p = 2 neither p*P - 1 = 1 nor p*P + 1 = 3 has a nontrivial square factor.
a(1) = 5 because for P = (product of the first prime) = 2, p = 5 is the least prime such that p*P - 1 = 9 = 3^2 is a square; for p = 3 none of p*P - 1 = 5 nor p*P + 1 = 7 has a nontrivial square factor.
a(2) = 29 because for P = (product of the first two primes) = 6, p = 29 is the least prime such that p*P + 1 = 5^2*7 has a square factor; for all primes 3 < p < 29 both of p*P +- 1 are squarefree.

Mathematica

Map[(k = 1; While[AllTrue[Prime[k] # + {-1, 1}, SquareFreeQ], k++]; Prime[k]) &, FoldList[Times, 1, Prime@ Range[24] ] ] (* Michael De Vlieger, Mar 28 2023 *)

Prog

(PARI) A361085(n, P=vecprod(primes(n)))=forprime(p=prime(n)+1, , (issquarefree(p*P-1)&&issquarefree(p*P+1))||return(p))

Crossrefs

Cf. A002110 (factorials), A013929 (numbers that are not squarefree), A007018, A228649 (both related).

Sequence in context: A016552 A321069 A355273 * A141578 A327748 A272345

Adjacent sequences: A361082 A361083 A361084 * A361086 A361087 A361088

Keyword

nonn,hard,more

Author

M. F. Hasler, Mar 28 2023

Extensions

a(30) from Michael S. Branicky, Mar 29 2023

a(31)-a(40) from Jinyuan Wang, Mar 30 2023

Status

approved