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 is a finite sequence having exactly 6 terms that are products, m*(m+1)*(m+2), where m-1 is one of the primes 7, 13, 19, 97, 349, 439.
Gaps between consecutive odd primes are even, so the number of composites within them is odd. Products of 5 or more consecutive integers cannot be prime-complete within a single prime gap: the Sylvester-Schur theorem guarantees a prime factor p > k (where k is the number of integers), forcing the inclusion of all primes <= p for prime-completeness. This requirement increases the necessary number of consecutive multipliers faster than prime gap sizes near m grow, making such configurations impossible or k >= 5. The only odd number between 1 and 5 is 3, hence all terms multiply exactly 3 consecutive composites.
This sequence is finite because the prime index of the greatest prime divisor of the product grows faster, after a point, than the number of distinct prime divisors of the product (those must be equal for the product to be prime-complete), and by reduction to Størmer's theorem (1897). For m*(m+1)*(m+2) to be prime-complete, all three integers must be p_k-smooth for some prime p_k. Consecutive p_k-smooth pairs (m, m+1) are finite and effectively computable; triple configurations (m, m+1, m+2 all smooth) are rarer still. Enumeration of all consecutive 17-smooth pairs finds the largest pair starts at m = 336140 (see A117581), well above the last term here at m = 440.
A Python program (link below) is provided to search intervals of m for the minimum difference, pi(gpf(m*(m+1)*(m+2))) - omega(m*(m+1)*(m+2)). It can be used to observe the growth of this minimum difference as m becomes large.
REFERENCES
C. 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, Python Program to Calculate N_3 Pi(GPF) - Omega
Paul Erdős, A theorem of Sylvester and Schur, J. London Math. Soc. 9 (1934), 282-288.
D. H. Lehmer, On a problem of Størmer, Illinois J. Math. 8 (1964), 57-79.
J. J. Sylvester, On arithmetical series, Messenger of Math. 21 (1892), part 1, 1-19 and part 2, 87-120.
Wikipedia, Størmer's theorem
EXAMPLE
a(1) = 8*9*10 = 720 (= 6!, that is 10!/7!, the unique nontrivial solution to a!*b!=c!) with factorization 2^4 * 3^2 * 5.
a(2) = 14*15*16 = 3360 with factorization 2^5 * 3 * 5 * 7.
a(3) = 20*21*22 = 9240 with factorization 2^3 * 3 * 5 * 7 * 11.
a(4) = 98*99*100 = 970200 with factorization 2^3 * 3^2 * 5^2 * 7^2 * 11.
a(5) = 350*351*352 = 43243200 with factorization 2^6 * 3^3 * 5^2 * 7 * 11 * 13.
a(6) = 440*441*442 = 85765680 with factorization 2^4 * 3^2 * 5 * 7^2 * 11 * 13 * 17.
MATHEMATICA
(* First, load function f from "Efficient Mathematica constructor program for A055932" at A376690, then: *)
Sort@ Select[Flatten@ f[12], And[#1 == Apply[Times, #2], NoneTrue[#2, PrimeQ], AllTrue[First[#2] + {-1, 3}, PrimeQ]] & @@ {#, Floor@ Surd[#, 3] + {0, 1, 2}} &] (* Michael De Vlieger, Feb 19 2026 *)
CROSSREFS
KEYWORD
nonn,fini,full
AUTHOR
Ken Clements, Jan 10 2026
STATUS
approved