OFFSET
1,1
COMMENTS
Let a number be designated "prime-complete" if, and only if, it has a complete contiguous set of prime divisors from 2 to its greatest prime divisor. The sequence, A055932, comprises all the prime-complete numbers, together with the number 1.
This sequence is the finite sequence of all prime-complete products of k consecutive composite integers. It comprises exactly 33 terms. There are 24 products with k=2 (subsequence of A141399 products), 8 with k=3 (subsequence of A391540 products), and 1 with k=4 (at 17297280). No prime-complete products exist for k >= 5. The greatest term is a(33) = 633555*633556 = 401392571580, as shown in A141399. This sequence is a supersequence of A391540.
Finiteness follows from an extension of Størmer's theorem: any k-tuple of consecutive p-smooth integers is bounded by the Størmer bound for consecutive p-smooth pairs. Combined with upper bounds on omega (count of distinct prime divisors) for products of k consecutive integers, this limits the search space to a finite, effectively computable region. The search space in A141399 for k=2 ran to a product of integers starting at 2.29*10^25, well beyond any Størmer bound for any value of k>=2.
This sequence is particularly significant because it limits the possible solutions to the Suranyi-Hickerson Conjecture (see Hajdu et al.) to the exact known:
6!*7! = 10! so 10!/7! = 6! = 8*9*10 = 720
3!*5!*7! = 10! so 10!/7! = 3!*5! = 8*9*10 = 720
2!*5!*14! = 16! so 16!/14! = 2!*5! = 15*16 = 240
2!*3!*3!*7! = 9! so 9!/7! = 2!*3!*3! = 8*9 = 72.
No other numbers in this sequence are factorials, or products of factorials, so no other solutions to Suranyi-Hickerson can exist.
REFERENCES
L. Hajdu, Á. Papp, T. Szakács, On the equation A!B!=C!, Journal of Number Theory 187 (2018), 160-165.
Florian Luca, On factorials which are products of factorials, Mathematical Proceedings of the Cambridge Philosophical Society, 143(3), 533-542 (2007).
Carl Størmer, Quelques théorèmes sur l'équation de Pell x^2 - Dy^2 = +-1 et leurs applications, Videnskabsselskabets Skrifter, I. Mat.-naturv. Klasse, No. 2 (1897).
LINKS
Ken Clements, Table of n, a(n) for n = 1..33
D. H. Lehmer, On a problem of Størmer, Illinois J. Math. 8 (1964), 57-79.
Wikipedia, Størmer's theorem.
EXAMPLE
a(1) = 8*9 = 72 = 2^3*3^2
a(2) = 9*10 = 90 = 2*3^2*5
a(3) = 14*15 = 210 = 2*3*5*7
a(4) = 15*16 = 240 = 2^4*3*5
a(5) = 20*21 = 420 = 2^2*3*5*7
a(6) = 24*25 = 600 = 2^3*3*5^2
a(7) = 8*9*10 = 720 = 2^4*3^2*5
PROG
(PARI)
is_prime_complete(m) = my(f=factor(m)[, 1]); #f == primepi(f[#f]);
a(max_p) = {
my(res = List());
forprime(p=7, max_p,
my(q=nextprime(p+1));
for(n=p+1, q-2,
my(m=n);
for(nxt=n+1, q-1,
m *= nxt;
if(is_prime_complete(m), listput(res, m))
)
)
);
vecsort(Vec(res))
}
print(a(700000));
CROSSREFS
KEYWORD
nonn,fini,full
AUTHOR
Ken Clements, Jan 30 2026
STATUS
approved