OFFSET
1,1
COMMENTS
There is no 5 in the prime factorization of these terms.
No other terms less than 10^14. - Michael S. Branicky, Feb 02 2023
1.69 * 10^15 < a(10) <= 7722272777722272. - Daniel Suteu, Feb 05 2023
REFERENCES
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, 1997, page 144, entry 567.
LINKS
Cliff Pickover et al., Exclusionary Squares and Cubes, rec.puzzles topic on google groups, January 2002.
Eric Weisstein's World of Mathematics, Distinct Prime Factors.
FORMULA
Assuming a(n) exists, a(n) >= A002110(n+1)/5 >> exp((1 + o(1)) * n * log(n)). (The inequality is presumably strict for all n; for n > 34 it seems that all A002110(n) are pandigital.) - Charles R Greathouse IV, Feb 05 2023
EXAMPLE
84 = 2^2 * 3 * 7 is the smallest integer with 3 distinct prime factors that is also an exclusionary square, because 84^2 = 7056, so a(3) = 84.
858 = 2 * 3 * 11 * 13 is the smallest integer with 4 distinct prime factors that is also an exclusionary square, because 858^2 = 736164, so a(4) = 858.
PROG
(PARI)
omega_exclusionary_squares(A, B, n) = A=max(A, vecprod(primes(n))); (f(m, p, j) = my(list=List()); forprime(q=p, sqrtnint(B\m, j), if(q == 5, next); my(v=m*q); while(v <= B, if(j==1, if(v>=A && #setintersect(Set(digits(v)), Set(digits(v^2))) == 0, listput(list, v)), if(v*(q+1) <= B, list=concat(list, f(v, q+1, j-1)))); v *= q)); list); vecsort(Vec(f(1, 2, n)));
a(n) = my(x=vecprod(primes(n)), y=2*x); while(1, my(v=omega_exclusionary_squares(x, y, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Feb 05 2023
CROSSREFS
KEYWORD
nonn,base,more
AUTHOR
Bernard Schott, Feb 02 2023
EXTENSIONS
a(4)-a(7) from Amiram Eldar, Feb 02 2023
a(8)-a(9) from Michael S. Branicky, Feb 02 2023
a(10) from Michael S. Branicky, Feb 07 2023
STATUS
approved