A389506 Smallest even number which cannot be written as the difference of two numbers which are less than and relatively prime to (the primorial) p(n)#.

2, 14, 194, 2282, 30002, 510470, 9699634, 223092806, 6469693154

Offset

2,1

Comments

Based on but not identical to Erdős Problem 854 (see link).

Links

Table of n, a(n) for n=2..10.

Thomas Bloom, Problem 854, Erdős Problems.

Example

For n=2, a(2)=2 since the only numbers (bounded by 6) relative prime with p(2)# = 3# = 6 are 1 and 5, and 2 is not the difference of them.
For n=3, a(3)=14, since the numbers relatively prime to and bounded by p(3)# = 30 are in the set [1, 7, 11, 13, 17, 19, 23, 29]. Now 19-17=2, 17-13=4, 23-17=6, 19-11=8, 29-19=10 and 29-17=12, but no two elements differ by 14 (nor 20, 24 or 26).

Prog

(SageMath)
def a(n):
    N=prod(Primes()[0:n]);
    R=[a for a in range(1, N) if gcd(N, a)==1];
    D=[j-i for (i, j) in Combinations(R, 2)];
    return min([i for i in range(2, N, 2) if i not in D]);
(PARI) a(n) = my(P=vecprod(primes(n)), v=select(x->(gcd(x, P)==1), [1..P]), list=List(), vs=[]); for (i=1, #v, for (j=i+1, #v, my(x=v[j]-v[i]); if (!vecsearch(vs, x), listput(list, x); vs = vecsort(list)); ); ); my(s=Set(list)); for (k=1, vecmax(v), if (!setsearch(s, 2*k), return(2*k)); ); \\ Michel Marcus, Oct 09 2025

Crossrefs

Cf. A051271, A051272, A002110, A389507, A007310, A007775, A008364-A008366, A166061, A166063.

Sequence in context: A381478 A268005 A158102 * A067902 A132611 A156327

Adjacent sequences: A389503 A389504 A389505 * A389507 A389508 A389509

Keyword

nonn,hard,more

Author

Stijn Cambie, Oct 07 2025

Extensions

a(7) from Michel Marcus, Oct 10 2025

a(8)-a(10) from Sean A. Irvine, Oct 14 2025

Status

approved