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.

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, 22, 23, 24, 19, 14, 1, 22, 20, 23, 24, 31, 27, 25, 26, 36, 28, 37, 29, 30, 42, 37, 22, 32, 37, 38, 35, 41, 36, 37, 43, 42, 37, 44, 44, 34, 33, 47, 1, 48, 49, 43, 53

Offset: 0

Author

Gordon Hamilton, Mar 31 2015

Keywords

• nonn
• easy

Comments

The iteration that leads to this sequence is worthy of consideration for the grade 2 classroom learning addition.

a(2^k)=1 for all nonnegative integers k as can be seen from A138591.

Examples

			a(23) = 23 because there are 23 numbers generated by the iteration:
                  23
                  /\
                 /  \
                /    \
               /      \
              /        \
             /          \
            /            \
          11             12
          /\             /|\
         /  \           / | \
        /    \         /  |  \
       /      \       3   4   5
      /        \     / \     / \
     5          6   1   2   2   3
    / \        /|\             / \
   2   3      / | \           1   2
      / \    /  |  \
     1   2  1   2   3
                   / \
                  1   2
a(24) = 13 because there are 13 numbers generated by the iteration:
          24
          /|\
         / | \
        /  |  \
       7   8   9
      / \     /|\
     3   4   / | \
    / \     /  |  \
   1   2   2   3   4
              / \
             1   2
		

Links

• Martin Büttner, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A138591.

Programs

Mathematica

fission[0] = 0;
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)]]];
fission /@ Range[0, 100] (* Martin Büttner, Jun 04 2015 *)

Extensions

Corrected and extended by Martin Büttner, Jun 04 2015