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A322583
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a(n) is the number of factorizations of n into factorial numbers larger than one; a(1) = 1.
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12
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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, 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, 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
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OFFSET
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1,24
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COMMENTS
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Values 1..7 occur for the first time at n = 1, 24, 576, 13824, 69120, 414720, 1658880.
In range 1..69120 differs from A034876 only at positions n = 1, 2, 9, 10 and 16.
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LINKS
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FORMULA
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EXAMPLE
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a(4) = 1 because 4 = 2! * 2!.
a(24) = 2 because 24 = 4! = 3! * 2! * 2!.
a(576) = 3 because 576 = 4! * 4! = 4! * 3! * 2! * 2! = (3!)^2 * (2!)^4.
a(13824) = 4 because 13824 = (4!)^3 = (4!)^2 * 3! * (2!)^2 = 4! * (3!)^2 * (2!)^4 = (3!)^3 * (2!)^6.
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.
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PROG
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(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))));
(PARI)
A034968(n) = { my(s=0, b=2, d); while(n, d = (n%b); s += d; n = (n-d)/b; b++); (s); };
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));
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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