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%I #43 Jan 17 2023 16:31:55
%S 24,576,720,2880,13824,17280,40320,69120,241920,331776,362880,414720,
%T 518400,725760,967680,1451520,1658880,2073600,2903040,3628800,5806080,
%U 7962624,8294400,8709120,9953280,12441600,14515200,17418240,23224320,29030400,34836480,39813120,43545600
%N Numbers m > 1 such that for all k > 1, m can be written as a product of factorials without using k!.
%C The name makes this different from "Numbers that are a product of factorials" (A001013). For example, numbers that can be written as a product of factorials > 1 in exactly one way are excluded as it is impossible to write such a product without using any of the factorials in this factorization. See the exclusion of 12 in the example section.
%C This is a primitive sequence related to A359750. A359750(n) = a(k) * A001013(m) for at least one pair (k, m).
%H David A. Corneth, <a href="/A359751/b359751.txt">Table of n, a(n) for n = 1..2908</a> (terms <= 10^24)
%e 2880 is in the sequence via 2880 = (2!)^2 * 6! = 4!*5! = (2!)^2 * 3! * 5!. The factorials > 1 that are factors in a least one of these products are 2!, 3!, 4!, 5!, 6!. None of these factorials occur as factors in all of these products. For example, 2! no factor in 4!*5!, 3! no factor in 4!*5!, 4! no factor in (2!)^2 * 6!, 5! no factor in (2!)^2 * 6!, 6! no factor in 4!*5!.
%e 24 is in the sequence (even though it is a factorial number) as 24 = 2! * 2! * 3! = 4!. So 24 can be written as a product of factorials in at least two ways (some of the factorials {2!, 3!, 4!}). But none of these factorials is in every factorization.
%e 48 is NOT in the sequence as 48 = 2! * 2! * 2! * 3! = 2! * 4!. So 48 can be written as a product of factorials in at least two ways (some of the factorials {2!, 3!, 4!}). But 2! is a factor of every factorization.
%e 12 is NOT in the sequence even though it can be written as a product of factorials, namely 2! * 3! = 12. As this is the only way to write 12 as a product of factorials, it is impossible to write 12 as a product of factorials without using 2!.
%o (PARI) is(n) = { if(n == 1, return(0)); my(i, factorials, e, res, v); f = factor(n); if(prime(#f~) != f[#f~, 1], return(0); ); if(f[,2] != vecsort(f[,2],,4), return(0); ); factorials = List(); e = List(); res = List(); for(i = 2, oo, v = valuation(n, i!); if(v > 0, listput(factorials, i!); listput(e, v); , break ) ); forvec(x = vector(#e-1, i, [0, e[i+1]]), c = prod(i = 1, #e-1, factorials[i+1]^x[i]); if(c <= n && denominator(n/c) == 1&& 1 << logint(n/c, 2) == n/c, listput(res, concat([valuation(n/c, 2)], x)) ) ); for(i = 1, #e, p = 1; for(j = 1, #res, p*=res[j][i]; ); if(p != 0, return(0) ) ); 1 }
%Y Cf. A001013, A359750.
%K nonn
%O 1,1
%A _David A. Corneth_ and _Peter Munn_, Jan 13 2023
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