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A266344
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a(n) = number of ways n can be divided into two factors that have the same number of digits in factorial base representation (the two different orders for unequal factors are counted only once).
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4
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1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 2, 0, 0, 0, 1, 0, 2, 1, 0, 0, 0, 0, 2, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 2, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 1, 3
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listen;
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OFFSET
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1,72
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LINKS
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FORMULA
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(In the above formula [ ] stands for Iverson bracket, giving as its result 1 only if d is less than or equal to n/d and in factorial base representation d and n/d require equal number of digits, and 0 otherwise.)
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EXAMPLE
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1 can be factored just one way, as 1*1, and thus a(1) = 1.
4 can be factored as 2*2, and thus also a(4) = 1, and generally for all perfect squares k, a(k) >= 1.
14 can be factored as 2*7, but as A007623(2) = 2 and A007623(7) = 101, with different number of digits in factorial base (and 1*14 fares even less well), a(14) = 0.
72 can be factored to two divisors so that the factorial base representations are of equal length as 6*12 or 8*9 (where the corresponding factorial base representations are "100" * "200" and "110" * "111"), thus a(72) = 2.
120 can be similarly factored as 6*20 ("100" * "310"), 8*15 ("110" * "211") and 10*12 ("120" * "200"), thus a(120) = 3.
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MATHEMATICA
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r = Most@ Reverse@ Range@ 10; Map[Length, Table[Flatten@ Map[Differences@ IntegerLength[#, MixedRadix@ r] &, Transpose@ {#, n/#}] &@ TakeWhile[Divisors@ n, # <= Sqrt@ n &], {n, 120}] /. k_ /; k > 0 -> Nothing] (* Michael De Vlieger, Dec 30 2015, Version 10.2 *)
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PROG
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(PARI)
A084558(n) = { my(m=1); if(0==n, n, while(m!<=n, m++); return(m-1)); }
for(n=1, 14161, write("b266344.txt", n, " ", A266344(n)));
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CROSSREFS
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Cf. A266345 (positions of records).
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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