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A241918 Table of partitions where the ordering is based on the modified partial sums of the exponents of primes in the prime factorization of n.

%I A241918 #35 Jul 09 2025 04:39:07
%S A241918 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,
%T A241918 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,
%U A241918 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
%N A241918 Table of partitions where the ordering is based on the modified partial sums of the exponents of primes in the prime factorization of n.
%C A241918 a(1) = 0 by convention (stands for an empty partition).
%C A241918 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.
%H A241918 Antti Karttunen, <a href="/A241918/b241918.txt">Table of n, a(n) for n = 1..10081; rows 1..521 flattened.</a>
%H A241918 <a href="http://oeis.org/wiki/User:Antti_Karttunen/Mail_by_Marc_LeBrun_Re_Partition_ordering_on_SeqFan_list_posted_12_Jan_2006">Marc LeBrun's original "crazy order" mapping for partitions</a> (Copy of Marc's Jan 11 2006 message in OEIS Wiki)
%F A241918 If A241914(n)=0 and A241914(n+1)=0, a(n) = A067255(n); otherwise, if A241914(n)=0 and A241914(n+1)>0, a(n) = A067255(n)+1; otherwise, if A241914(n)>0 and A241914(n+1)=0, a(n) = a(n-1) + A067255(n) - 1, otherwise, when A241914(n)>0 and A241914(n+1)>0, a(n) = a(n-1) + A067255(n).
%e A241918 Table begins:
%e A241918 Row     Partition
%e A241918 [ 1]    0;         (stands for empty partition)
%e A241918 [ 2]    1;         (as 2 = 2^1)
%e A241918 [ 3]    1,1;       (as 3 = 2^0 * 3^1)
%e A241918 [ 4]    2;         (as 4 = 2^2)
%e A241918 [ 5]    1,1,1;     (as 5 = 2^0 * 3^0 * 5^1)
%e A241918 [ 6]    2,2;       (as 6 = 2^1 * 3^1)
%e A241918 [ 7]    1,1,1,1;   (as 7 = 2^0 * 3^0 * 5^0 * 7^1)
%e A241918 [ 8]    3;         (as 8 = 2^3)
%e A241918 [ 9]    1,2;       (as 9 = 2^0 * 3^2)
%e A241918 [10]    2,2,2;     (as 10 = 2^1 * 3^0 * 5^1)
%e A241918 [11]    1,1,1,1,1;
%e A241918 [12]    3,3;
%e A241918 [13]    1,1,1,1,1,1;
%e A241918 [14]    2,2,2,2;
%e A241918 [15]    1,2,2;     (as 15 = 2^0 * 3^1 * 5^1)
%e A241918 [16]    4;
%e A241918 [17]    1,1,1,1,1,1,1;
%e A241918 [18]    2,3;       (as 18 = 2^1 * 3^2)
%e A241918 etc.
%e A241918 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.
%t A241918 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 *)
%o A241918 (Scheme, with _Antti Karttunen_'s IntSeq-library)
%o A241918 (definec (A241918 n) (cond ((zero? (A241914 n)) (if (zero? (A241914 (+ n 1))) (A067255 n) (+ 1 (A067255 n)))) ((zero? (A241914 (+ 1 n))) (+ (A241918 (- n 1)) (- (A067255 n) 1))) (else (+ (A241918 (- n 1)) (A067255 n)))))
%Y A241918 For n>=2, the length of row n is given by A061395(n).
%Y A241918 Cf. also A067255, A203623, A241914.
%Y A241918 Other tables of partitions: A112798 (also based on prime factorization), A227739, A242628 (encoded in the binary representation of n), and A036036-A036037, A080576-A080577, A193073 for various lexicographical orderings.
%Y A241918 Permutation A241909 maps between order of partitions employed here, and the order employed in A112798.
%Y A241918 Permutation A122111 is induced when partitions in this list are conjugated.
%Y A241918 A241912 gives the row numbers for which the corresponding rows in A112798 and here are the conjugate partitions of each other.
%K A241918 nonn,tabf
%O A241918 1,5
%A A241918 _Antti Karttunen_, May 03 2014, based on _Marc LeBrun_'s Jan 11 2006 message on SeqFan mailing list