A360301 - OEIS

%I #32 Feb 12 2023 05:38:15

%S 2,18,84,858,31122,3383898,188841114,68588585868,440400004044,

%T 7722272777722272

%N Smallest exclusionary square (A029783) with exactly n distinct prime factors.

%C There is no 5 in the prime factorization of these terms.

%C No other terms less than 10^14. - _Michael S. Branicky_, Feb 02 2023

%C 1.69 * 10^15 < a(10) <= 7722272777722272. - _Daniel Suteu_, Feb 05 2023

%D Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.

%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, 1997, page 144, entry 567.

%H Cliff Pickover et al., <a href="https://groups.google.com/forum/#!topic/rec.puzzles/ubSItPD_DGY">Exclusionary Squares and Cubes</a>, rec.puzzles topic on google groups, January 2002.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DistinctPrimeFactors.html">Distinct Prime Factors</a>.

%F 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

%e 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.

%e 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.

%o (PARI)

%o 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)));

%o 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

%Y Cf. A029783.

%Y Similar: A060319 (Fibonacci), A083002 (oblong), A359960 (Niven), A359961 (Zuckerman).

%K nonn,base,more

%O 1,1

%A _Bernard Schott_, Feb 02 2023

%E a(4)-a(7) from _Amiram Eldar_, Feb 02 2023

%E a(8)-a(9) from _Michael S. Branicky_, Feb 02 2023

%E a(10) from _Michael S. Branicky_, Feb 07 2023