cp's OEIS Frontend

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).

%I A361085 #33 Apr 01 2023 23:27:28
%S A361085 3,5,29,31,139,167,43,127,211,41,607,1223,71,769,1549,947,269,1129,
%T A361085 163,577,673,107,4057,1979,433,3833,4177,383,1723,409,2399,4517,3803,
%U A361085 3061,3299,457,3779,971,5749,2843,13709
%N 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).
%C A361085 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.)
%C A361085 See also Zumkeller's contested comment in A007018 and the discussion in the linked MathOverflow page.
%H A361085 Dan Asimov, <a href="https://mailman.xmission.com/hyperkitty/list/math-fun@mailman.xmission.com/thread/5TNAYAUATD2D5IHWQQIIBPF3QE2O3ILR/">Interesting sequence on MathOverflow</a>, math-fun mailing list, Mar 28 2023.
%H A361085 Fredrick M. Nelson, <a href="https://mathoverflow.net/questions/443446/">Does a(0)=6, a(n+1)=a(n)^3-a(n), define a square-free sequence?</a>, MathOverflow, Mar 24 2023.
%e A361085 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.
%e A361085 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.
%e A361085 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.
%t A361085 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 *)
%o A361085 (PARI) A361085(n, P=vecprod(primes(n)))=forprime(p=prime(n)+1,,(issquarefree(p*P-1)&&issquarefree(p*P+1))||return(p))
%Y A361085 Cf. A002110 (factorials), A013929 (numbers that are not squarefree), A007018, A228649 (both related).
%K A361085 nonn,hard,more
%O A361085 0,1
%A A361085 _M. F. Hasler_, Mar 28 2023
%E A361085 a(30) from _Michael S. Branicky_, Mar 29 2023
%E A361085 a(31)-a(40) from _Jinyuan Wang_, Mar 30 2023