A359750 Numbers that are a product of one or more factorials j!, j >= 2, in at least two ways.
24, 48, 96, 144, 192, 288, 384, 576, 720, 768, 864, 1152, 1440, 1536, 1728, 2304, 2880, 3072, 3456, 4320, 4608, 5184, 5760, 6144, 6912, 8640, 9216, 10368, 11520, 12288, 13824, 17280, 18432, 20736, 23040, 24576, 25920, 27648, 31104, 34560, 36864, 40320, 41472, 46080
Offset: 1
Keywords
Examples
24 = 2! * 3! = 4!. 144 = (2!)^2 * (3!)^2 = 3! * 4!.
Crossrefs
Cf. A001013.
Programs
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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)) ) ); #res >= 2 } \\ David A. Corneth, Jan 13 2023