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A241918
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Table of partitions where the ordering is based on the modified partial sums of the exponents of primes in the prime factorization of n.
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15
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0, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 4, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5
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OFFSET
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1,5
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COMMENTS
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a(1) = 0 by convention (stands for an empty partition).
For n >= 2, A203623(n-1)+2 gives the index to the beginning of row n and for n>=1, A203623(n)+1 is the index to the end of row n.
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LINKS
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FORMULA
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EXAMPLE
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Table begins:
Row Partition
[ 1] 0; (stands for empty partition)
[ 2] 1; (as 2 = 2^1)
[ 3] 1,1; (as 3 = 2^0 * 3^1)
[ 4] 2; (as 4 = 2^2)
[ 5] 1,1,1; (as 5 = 2^0 * 3^0 * 5^1)
[ 6] 2,2; (as 6 = 2^1 * 3^1)
[ 7] 1,1,1,1; (as 7 = 2^0 * 3^0 * 5^0 * 7^1)
[ 8] 3; (as 8 = 2^3)
[ 9] 1,2; (as 9 = 2^0 * 3^2)
[10] 2,2,2; (as 10 = 2^1 * 3^0 * 5^1)
[11] 1,1,1,1,1;
[12] 3,3;
[13] 1,1,1,1,1,1;
[14] 2,2,2,2;
[15] 1,2,2; (as 15 = 2^0 * 3^1 * 5^1)
[16] 4;
[17] 1,1,1,1,1,1,1;
[18] 2,3; (as 18 = 2^1 * 3^2)
etc.
If n is 2^k (k>=1), then the partition is a singleton {k}, otherwise, add one to the exponent of 2 (= A007814(n)), and subtract one from the exponent of the greatest prime dividing n (= A071178(n)), leaving the intermediate exponents as they are, and then take partial sums of all, thus resulting for e.g. 15 = 2^0 * 3^1 * 5^1 the modified sequence of exponents {0+1, 1, 1-1} -> {1,1,0}, whose partial sums {1,1+1,1+1+0} -> {1,2,2} give the corresponding partition at row 15.
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MATHEMATICA
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Table[If[n == 1, {0}, Function[s, Function[t, Accumulate[If[Length@ t < 2, {0}, Join[{1}, ConstantArray[0, Length@ t - 2], {-1}]] + ReplacePart[t, Map[#1 -> #2 & @@ # &, s]]]]@ ConstantArray[0, Transpose[s][[1, -1]]]][FactorInteger[n] /. {p_, e_} /; p > 0 :> {PrimePi@ p, e}]], {n, 31}] // Flatten (* Michael De Vlieger, May 12 2017 *)
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PROG
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CROSSREFS
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For n>=2, the length of row n is given by A061395(n).
Permutation A241909 maps between order of partitions employed here, and the order employed in A112798.
Permutation A122111 is induced when partitions in this list are conjugated.
A241912 gives the row numbers for which the corresponding rows in A112798 and here are the conjugate partitions of each other.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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