This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A305188 #105 Jul 23 2020 03:27:17 %S A305188 6,10,12,15,20,21,24,28,30,35,36,42,45,55,56,60,66,70,72,78,84,90,91, %T A305188 105,110,120,126,132,136,140,153,156,165,168,171,180,182,190,210,220, %U A305188 231,240,252,253,272,276,280,286,300,306,325,330,336,342,351,360,364 %N A305188 Numbers that are equal to a nontrivial multinomial coefficient (i.e., equal to k!/(k1!*...*km!) with k1 + ... + km = k, k-2 >= k1 >= ... >= km). %C A305188 This sequence answers the following question: what numbers correspond to the number of permutations of a number of items that is lower than the number of permutations itself? It means that the underlying structure has some form of redundance / symmetry. %C A305188 It can be shown that: %C A305188 - no prime number is part of this sequence (see A304938) %C A305188 - some nonprimes are not part of the sequence (beginning with 1, 4, 8, 9, 14, 16, 18, ...) %C A305188 - any number that is a factorial of another integer e is part of this sequence (k=e, k1=ki=...=ke=1). %C A305188 This sequence is a generalization of A006987. %C A305188 From _David A. Corneth_, May 28 2018: (Start) %C A305188 Also the numbers that are the number of permutations of either: %C A305188 - sets of balls with two distinct colors of balls where each color occurs at least twice; %C A305188 - sets of balls with at least three distinct colors of balls. %C A305188 (End) %C A305188 From _Amiram Eldar_, Jul 23 2020: (Start) %C A305188 The asymptotic density of this sequence is 0 (Niven, 1951). %C A305188 The number of terms not exceeding x is (1 + sqrt(2)) * x^(1/2) + o(x^(1/2)) (Erdős, 1954). (End) %H A305188 David A. Corneth, <a href="/A305188/b305188.txt">Table of n, a(n) for n = 1..10000</a> %H A305188 Vincent Champain, <a href="https://repl.it/@vcch/OEIS305188">Calculation of the terms of the sequence including a k1..km list so that a(n) = k!/(k1!*...*km!) with k1+...+km = k and m > 1</a>, 2018. %H A305188 Vincent Champain, <a href="/A305188/a305188.txt">Python program for A305188</a> %H A305188 David A. Corneth, <a href="/A305188/a305188.gp.txt">Terms with the corresponding tuples. </a> %H A305188 Paul Erdős, <a href="http://www.jstor.org/stable/2306893">The Number of Multinomial Coefficients</a>, The American Mathematical Monthly, Vol. 61, No. 1 (1954), pp. 37-39. %H A305188 Ivan Niven, <a href="https://doi.org/10.1090/S0002-9904-1951-09543-9">The asymptotic density of sequences</a>, Bull. Amer. Math. Soc., Vol. 57 (1951), pp. 420-434. See theorem 2, p. 428. %e A305188 a(1) = 6 because all numbers lower than 6 are either prime or a power of primes. %e A305188 105 is a term of the sequence because 105 is equal to a multinomial coefficient: 105 = (4+2+1)! / (4! * 2! * 1!) and 105 is the number of ways 7 balls can be sorted where 4 are red, 2 are yellow and one is blue. %e A305188 2016 is a term because 64! / (62! * 2!) = 2016. - _David A. Corneth_, May 29 2018 %t A305188 mult[w_] := Total[w]!/Times @@ (w!); L = {}; Do[ t = mult /@ Select[ IntegerPartitions@ n, #[[1]] < n-1 &]; L = Union[L, Select[t, # <= 400 &]], {n, 3, 30}]; L (* Terms < 400, _Giovanni Resta_, May 27 2018 *) %o A305188 (Python) # see link above %Y A305188 Cf. A008480, A304938, A006987. %K A305188 nonn %O A305188 1,1 %A A305188 _Vincent Champain_, May 27 2018 %E A305188 a(28)-a(57) from _Giovanni Resta_, May 27 2018