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A256504 Summative Fission - For a positive integer n, find the greatest number of consecutive positive integers (at least 2) which add to n. For each of these do the same ... iterate to completion. a(n) = the total number of integers (including n itself) defined.

%I A256504 #13 Jun 05 2015 03:31:33
%S A256504 0,1,1,3,1,5,6,5,1,6,7,12,10,12,11,12,1,8,16,14,17,18,18,23,13,21,18,
%T A256504 22,23,24,19,14,1,22,20,23,24,31,27,25,26,36,28,37,29,30,42,37,22,32,
%U A256504 37,38,35,41,36,37,43,42,37,44,44,34,33,47,1,48,49,43,53
%N A256504 Summative Fission - For a positive integer n, find the greatest number of consecutive positive integers (at least 2) which add to n. For each of these do the same ... iterate to completion. a(n) = the total number of integers (including n itself) defined.
%C A256504 The iteration that leads to this sequence is worthy of consideration for the grade 2 classroom learning addition.
%C A256504 a(2^k)=1 for all nonnegative integers k as can be seen from A138591.
%H A256504 Martin Büttner, <a href="/A256504/b256504.txt">Table of n, a(n) for n = 0..10000</a>
%e A256504 a(23) = 23 because there are 23 numbers generated by the iteration:
%e A256504                   23
%e A256504                   /\
%e A256504                  /  \
%e A256504                 /    \
%e A256504                /      \
%e A256504               /        \
%e A256504              /          \
%e A256504             /            \
%e A256504           11             12
%e A256504           /\             /|\
%e A256504          /  \           / | \
%e A256504         /    \         /  |  \
%e A256504        /      \       3   4   5
%e A256504       /        \     / \     / \
%e A256504      5          6   1   2   2   3
%e A256504     / \        /|\             / \
%e A256504    2   3      / | \           1   2
%e A256504       / \    /  |  \
%e A256504      1   2  1   2   3
%e A256504                    / \
%e A256504                   1   2
%e A256504 a(24) = 13 because there are 13 numbers generated by the iteration:
%e A256504           24
%e A256504           /|\
%e A256504          / | \
%e A256504         /  |  \
%e A256504        7   8   9
%e A256504       / \     /|\
%e A256504      3   4   / | \
%e A256504     / \     /  |  \
%e A256504    1   2   2   3   4
%e A256504               / \
%e A256504              1   2
%t A256504 fission[0] = 0;
%t A256504 fission[n_] := fission@n = Module[{div = SelectFirst[Reverse@Divisors[2 n], (OddQ@# == IntegerQ[n/#] && n/# > (# - 1)/2) &]}, If[div == 1, 1, 1 + Total[fission /@ (Range@div + n/div - (div + 1)/2)]]];
%t A256504 fission /@ Range[0, 100] (* _Martin Büttner_, Jun 04 2015 *)
%Y A256504 Cf. A138591.
%K A256504 nonn,easy
%O A256504 0,4
%A A256504 _Gordon Hamilton_, Mar 31 2015
%E A256504 Corrected and extended by _Martin Büttner_, Jun 04 2015