A360301 Smallest exclusionary square (A029783) with exactly n distinct prime factors.

2, 18, 84, 858, 31122, 3383898, 188841114, 68588585868, 440400004044, 7722272777722272

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

Table of n, a(n) for n=1..10.

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

Cf. A029783.

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

Sequence in context: A091940 A068605 A343514 * A070171 A357757 A172529

Adjacent sequences: A360298 A360299 A360300 * A360302 A360303 A360304

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