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A322583 a(n) is the number of factorizations of n into factorial numbers larger than one; a(1) = 1.

%I A322583 #24 Dec 30 2018 00:04:34
%S A322583 1,1,0,1,0,1,0,1,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,
%T A322583 0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,
%U A322583 0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1
%N A322583 a(n) is the number of factorizations of n into factorial numbers larger than one; a(1) = 1.
%C A322583 Values 1..7 occur for the first time at n = 1, 24, 576, 13824, 69120, 414720, 1658880.
%C A322583 In range 1..69120 differs from A034876 only at positions n = 1, 2, 9, 10 and 16.
%H A322583 Antti Karttunen, <a href="/A322583/b322583.txt">Table of n, a(n) for n = 1..69120</a>
%H A322583 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A322583 a(24^n) = a(A009968(n)) = n+1.
%e A322583 a(4) = 1 because 4 = 2! * 2!.
%e A322583 a(24) = 2 because 24 = 4! = 3! * 2! * 2!.
%e A322583 a(576) = 3 because 576 = 4! * 4! = 4! * 3! * 2! * 2! = (3!)^2 * (2!)^4.
%e A322583 a(13824) = 4 because 13824 = (4!)^3 = (4!)^2 * 3! * (2!)^2 = 4! * (3!)^2 * (2!)^4 = (3!)^3 * (2!)^6.
%e A322583 a(69120) = 5 because 69120 = 6! * 4! * 2! * 2! = 6! * 3! * 2! * 2! * 2! * 2! = 5! * 3! * 3! * 2! * 2! * 2! * 2! = 5! * 4! * 4! = 5! * 4! * 3! * 2! * 2!. Note that 69120 < 24^4 = 331776.
%o A322583 (PARI) A322583(n, m=n) = if(1==n, 1, my(s=0); for(i=2, oo, my(f=i!); if(f>m, return(s)); if(!(n%f), s += A322583(n/f, f))));
%o A322583 (PARI)
%o A322583 A034968(n) = { my(s=0, b=2, d); while(n, d = (n%b); s += d; n = (n-d)/b; b++); (s); };
%o A322583 A322583(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&(1==A034968(d)), s += A322583(n/d, d))); (s));
%Y A322583 Cf. A000142, A001013 (positions of nonzero terms), A009968, A034876, A034968.
%K A322583 nonn
%O A322583 1,24
%A A322583 _Antti Karttunen_, Dec 25 2018