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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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Last modified January 20 10:23 EST 2018. Contains 297960 sequences.