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# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a055881 Showing 1-1 of 1 %I A055881 #123 Jul 23 2022 05:53:42 %S A055881 1,2,1,2,1,3,1,2,1,2,1,3,1,2,1,2,1,3,1,2,1,2,1,4,1,2,1,2,1,3,1,2,1,2, %T A055881 1,3,1,2,1,2,1,3,1,2,1,2,1,4,1,2,1,2,1,3,1,2,1,2,1,3,1,2,1,2,1,3,1,2, %U A055881 1,2,1,4,1,2,1,2,1,3,1,2,1,2,1,3,1,2,1,2,1,3,1,2,1,2,1,4,1,2,1,2,1,3,1,2,1 %N A055881 a(n) = largest m such that m! divides n. %C A055881 Number of factorial divisors of n. - _Amarnath Murthy_, Oct 19 2002 %C A055881 The sequence may be constructed as follows. Step 1: start with 1, concatenate and add +1 to last term gives: 1,2. Step 2: 2 is the last term so concatenate twice those terms and add +1 to last term gives: 1, 2, 1, 2, 1, 3 we get 6 terms. Step 3: 3 is the last term, concatenate 3 times those 6 terms and add +1 to last term gives: 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, iterates. At k-th step we obtain (k+1)! terms. - _Benoit Cloitre_, Mar 11 2003 %C A055881 From _Benoit Cloitre_, Aug 17 2007, edited by _M. F. Hasler_, Jun 28 2016: %C A055881 Another way to construct the sequence: start from an infinite series of 1's: %C A055881 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... Replace every second 1 by a 2 giving: %C A055881 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, ... Replace every third 2 by a 3 giving: %C A055881 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, ... Replace every fourth 3 by a 4 etc. (End) %C A055881 This sequence is the fixed point, starting with 1, of the morphism m, where m(1) = 1, 2, and for k > 1, m(k) is the concatenation of m(k - 1), the sequence up to the first k, and k + 1. Thus m(2) = 1, 2, 1, 3; m(3) = 1, 2, 1, 3, 1, 2, 1, 2, 1, 4; m(4) = 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 5, etc. - _Franklin T. Adams-Watters_, Jun 10 2009 %C A055881 All permutations of n elements can be listed as follows: Start with the (arbitrary) permutation P(0), and to obtain P(n + 1), reverse the first a(n) + 1 elements in P(n). The last permutation is the reversal of the first, so the path is a cycle in the underlying graph. See example and fxtbook link. - _Joerg Arndt_, Jul 16 2011 %C A055881 Positions of rightmost change with incrementing rising factorial numbers, see example. - _Joerg Arndt_, Dec 15 2012 %C A055881 Records appear at factorials. - _Robert G. Wilson v_, Dec 21 2012 %C A055881 One more than the number of trailing zeros (A230403(n)) in the factorial base representation of n (A007623(n)). - _Antti Karttunen_, Nov 18 2013 %C A055881 A062356(n) and a(n) coincide quite often. - _R. J. Cano_, Aug 04 2014 %C A055881 For n>0 and 1<=j<=(n+1)!-1, (n+1)^2-1=A005563(n) is the number of times that a(j)=n-1. - _R. J. Cano_, Dec 23 2016 %H A055881 Antti Karttunen, Table of n, a(n) for n = 1..10080 %H A055881 Joerg Arndt, Matters Computational (The Fxtbook), section 10.4, pp.245-248 (prefix reversals); section 10.5, pp. 248-250 (Heap's method). %H A055881 R. J. Cano, Alternative sequencer (PARI/GP). %H A055881 Claude Lenormand, Comments on this sequence. %H A055881 József Sándor, On Additive Analogues of Certain Arithmetic Smarandache Functions. %H A055881 Index entries for sequences related to factorial base representation. %F A055881 G.f.: Sum_{k > 0} x^(k!)/(1 - x^(k!)). - _Vladeta Jovovic_, Dec 13 2002 %F A055881 a(n) = A230403(n)+1. - _Antti Karttunen_, Nov 18 2013 %F A055881 a(n) = A230415(n-1,n) = A230415(n,n-1) = A230417(n,n-1). - _Antti Karttunen_, Nov 19 2013 %F A055881 a(m!+n) = a(n) if 1 <= n <= m*m! - 1 = A001563(m) - 1. - _R. J. Cano_, Jun 27 2016 %F A055881 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = e - 1 (A091131). - _Amiram Eldar_, Jul 23 2022 %e A055881 a(12) = 3 because 3! is highest factorial to divide 12. %e A055881 From _Joerg Arndt_, Jul 16 2011: (Start) %e A055881 All permutations of 4 elements via prefix reversals: %e A055881 n: permutation a(n)+1 %e A055881 0: [ 0 1 2 3 ] - %e A055881 1: [ 1 0 2 3 ] 2 %e A055881 2: [ 2 0 1 3 ] 3 %e A055881 3: [ 0 2 1 3 ] 2 %e A055881 4: [ 1 2 0 3 ] 3 %e A055881 5: [ 2 1 0 3 ] 2 %e A055881 6: [ 3 0 1 2 ] 4 %e A055881 7: [ 0 3 1 2 ] 2 %e A055881 8: [ 1 3 0 2 ] 3 %e A055881 9: [ 3 1 0 2 ] 2 %e A055881 10: [ 0 1 3 2 ] 3 %e A055881 11: [ 1 0 3 2 ] 2 %e A055881 12: [ 2 3 0 1 ] 4 %e A055881 13: [ 3 2 0 1 ] 2 %e A055881 14: [ 0 2 3 1 ] 3 %e A055881 15: [ 2 0 3 1 ] 2 %e A055881 16: [ 3 0 2 1 ] 3 %e A055881 17: [ 0 3 2 1 ] 2 %e A055881 18: [ 1 2 3 0 ] 4 %e A055881 19: [ 2 1 3 0 ] 2 %e A055881 20: [ 3 1 2 0 ] 3 %e A055881 21: [ 1 3 2 0 ] 2 %e A055881 22: [ 2 3 1 0 ] 3 %e A055881 23: [ 3 2 1 0 ] 2 %e A055881 (End) %e A055881 From _Joerg Arndt_, Dec 15 2012: (Start) %e A055881 The first few rising factorial numbers (dots for zeros) with 4 digits and the positions of the rightmost change with incrementing are: %e A055881 [ 0] [ . . . . ] - %e A055881 [ 1] [ 1 . . . ] 1 %e A055881 [ 2] [ . 1 . . ] 2 %e A055881 [ 3] [ 1 1 . . ] 1 %e A055881 [ 4] [ . 2 . . ] 2 %e A055881 [ 5] [ 1 2 . . ] 1 %e A055881 [ 6] [ . . 1 . ] 3 %e A055881 [ 7] [ 1 . 1 . ] 1 %e A055881 [ 8] [ . 1 1 . ] 2 %e A055881 [ 9] [ 1 1 1 . ] 1 %e A055881 [10] [ . 2 1 . ] 2 %e A055881 [11] [ 1 2 1 . ] 1 %e A055881 [12] [ . . 2 . ] 3 %e A055881 [13] [ 1 . 2 . ] 1 %e A055881 [14] [ . 1 2 . ] 2 %e A055881 [15] [ 1 1 2 . ] 1 %e A055881 [16] [ . 2 2 . ] 2 %e A055881 [17] [ 1 2 2 . ] 1 %e A055881 [18] [ . . 3 . ] 3 %e A055881 [19] [ 1 . 3 . ] 1 %e A055881 [20] [ . 1 3 . ] 2 %e A055881 [21] [ 1 1 3 . ] 1 %e A055881 [22] [ . 2 3 . ] 2 %e A055881 [23] [ 1 2 3 . ] 1 %e A055881 [24] [ . . . 1 ] 4 %e A055881 [25] [ 1 . . 1 ] 1 %e A055881 [26] [ . 1 . 1 ] 2 %e A055881 (End) %t A055881 Table[Length[Intersection[Divisors[n], Range[5]!]], {n, 125}] (* _Alonso del Arte_, Dec 10 2012 *) %t A055881 f[n_] := Block[{m = 1}, While[Mod[n, m!] == 0, m++]; m - 1]; Array[f, 105] (* _Robert G. Wilson v_, Dec 21 2012 *) %o A055881 (Scheme) %o A055881 (define (A055881 n) (let loop ((n n) (i 2)) (cond ((not (zero? (modulo n i))) (- i 1)) (else (loop (/ n i) (+ 1 i)))))) %o A055881 (PARI) See Cano link. %o A055881 (PARI) n=5; f=n!; x='x+O('x^f); Vec(sum(k=1,n,x^(k!)/(1-x^(k!)))) \\ _Joerg Arndt_, Jan 28 2014 %o A055881 (PARI) a(n)=for(k=2,n+1,if(n%k, return(k-1),n/=k)) \\ _Charles R Greathouse IV_, May 28 2015 %Y A055881 Cf. A055874, A055926, A055770, A062356, A073575, A091131, A230403, A230404, A230405, A076733, A232096, A232098, A233285, A233267, A233269, A231719, A232741, A232742, A232743, A232744, A232745, A060832 (partial sums). %Y A055881 This sequence occurs also in the next to middle diagonals of A230415 and as the second rightmost column of triangle A230417. %Y A055881 Other sequences related to factorial base representation (A007623): A034968, A084558, A099563, A060130, A227130, A227132, A227148, A227149, A153880. %Y A055881 Analogous sequence for binary (base-2) representation: A001511. %K A055881 easy,nonn %O A055881 1,2 %A A055881 _Leroy Quet_ and _Labos Elemer_, Jul 16 2000 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE