cp's OEIS Frontend

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.

%I A256774 #22 Apr 22 2015 12:04:05
%S A256774 1,2,3,4,6,9,16,24,25,64,120,125,216,625,720,1296,2401,5040,16807,
%T A256774 32768,40320,59049,262144,362880,531441,1000000,3628800,10000000,
%U A256774 19487171,39916800,214358881,429981696,479001600,815730721,5159780352,6227020800,10604499373,20661046784,87178291200,289254654976
%N 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.
%C A256774 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.
%C A256774 A000142 (without its first term that stands for 0!) is a subsequence.
%H A256774 Michael De Vlieger, <a href="/A256774/b256774.txt">Table of n, a(n) for n = 1..1346</a>
%e A256774 With n=1: 1! < 2! gives a(1)=1, a(2)=2.
%e A256774 With n=2: 2! < 3^1 < 2^2 < 3! gives a(3)=3, a(4)=4, a(5)=6.
%e A256774 With n=3: 3! < 3^2 < 4^2 < 4! gives a(6)=9, a(7)=16, a(8)=24.
%e A256774 With n=4: 4! < 5^2 < 4^3 < 5! gives a(9)=25, a(10)=64, a(11)=120.
%e A256774 With n=5: 5! < 5^3 < 6^3 < 5^4 < 6! gives a(12)=125, a(13)=216, a(14)=625, a(15)=720
%t A256774 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 *)
%o A256774 (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
%Y A256774 Cf. A000142, A039960, A060151, A074181, A074182, A074184, A111683.
%K A256774 nonn,tabf
%O A256774 1,2
%A A256774 _Juan Castaneda_, Apr 10 2015