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A359751
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Numbers m > 1 such that for all k > 1, m can be written as a product of factorials without using k!.
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3
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24, 576, 720, 2880, 13824, 17280, 40320, 69120, 241920, 331776, 362880, 414720, 518400, 725760, 967680, 1451520, 1658880, 2073600, 2903040, 3628800, 5806080, 7962624, 8294400, 8709120, 9953280, 12441600, 14515200, 17418240, 23224320, 29030400, 34836480, 39813120, 43545600
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OFFSET
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1,1
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COMMENTS
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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.
This is a primitive sequence related to A359750. A359750(n) = a(k) * A001013(m) for at least one pair (k, m).
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LINKS
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EXAMPLE
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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!.
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.
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.
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!.
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PROG
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(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 }
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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