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 A256774 All factorials n! along with powers of the numbers n and n+1 that fall in between n! and (n+1)!, in increasing order. 1
 1, 2, 3, 4, 6, 9, 16, 24, 25, 64, 120, 125, 216, 625, 720, 1296, 2401, 5040, 16807, 32768, 40320, 59049, 262144, 362880, 531441, 1000000, 3628800, 10000000, 19487171, 39916800, 214358881, 429981696, 479001600, 815730721, 5159780352, 6227020800, 10604499373, 20661046784, 87178291200, 289254654976 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS For each positive integer n, we consider the two factorials n! and (n+1)! as lower and upper bounds of an interval. Then we look for all powers of n and all powers of n+1 that fall inside that interval. We sort those numbers in increasing order, and we append them to the sequence without allowing duplicates. Then we move on to the next integer, and so on. A000142 (without its first term that stands for 0!) is a subsequence. LINKS Michael De Vlieger, Table of n, a(n) for n = 1..1346 EXAMPLE With n=1: 1! < 2! gives a(1)=1, a(2)=2. With n=2: 2! < 3^1 < 2^2 < 3! gives a(3)=3, a(4)=4, a(5)=6. With n=3: 3! < 3^2 < 4^2 < 4! gives a(6)=9, a(7)=16, a(8)=24. With n=4: 4! < 5^2 < 4^3 < 5! gives a(9)=25, a(10)=64, a(11)=120. With n=5: 5! < 5^3 < 6^3 < 5^4 < 6! gives a(12)=125, a(13)=216, a(14)=625, a(15)=720 MATHEMATICA f[n_] := Block[{a = n!, b = (n + 1)!}, Sort@ Union[{a}, n^Range[Ceiling@ Log[n, a], Floor@ Log[n, b]], (n + 1)^Range[Ceiling@ Log[n + 1, a], Floor@ Log[n + 1, b]]]]; {1}~Join~(f /@ Range[2, 14] // Flatten) (* Michael De Vlieger, Apr 15 2015 *) PROG (PARI) tabf(nn) = {print([1]); for (n=2, nn, v = [n!]; ka = ceil(log(n!+1)/log(n)); kb = floor(log((n+1)!-1)/log(n)); for (k=ka, kb, v = concat(v, n^k); ); ka = ceil(log(n!+1)/log(n+1)); kb = floor(log((n+1)!-1)/log(n+1)); for (k=ka, kb, v = concat(v, (n+1)^k); ); print(vecsort(v)); ); } \\ Michel Marcus, Apr 22 2015 CROSSREFS Cf. A000142, A039960, A060151, A074181, A074182, A074184, A111683. Sequence in context: A268710 A223900 A192267 * A213682 A103481 A215285 Adjacent sequences:  A256771 A256772 A256773 * A256775 A256776 A256777 KEYWORD nonn,tabf AUTHOR Juan Castaneda, Apr 10 2015 STATUS approved

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