A368822 Number of compositions of n into two non-refactorable parts.
0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 4, 3, 4, 5, 4, 6, 8, 8, 8, 11, 10, 9, 10, 12, 12, 15, 14, 13, 14, 17, 16, 19, 20, 19, 20, 24, 22, 23, 22, 23, 24, 27, 24, 27, 28, 27, 28, 34, 32, 31, 32, 35, 34, 37, 36, 37, 38, 41, 38, 41, 40, 41, 40, 45, 44, 43, 44, 49, 48, 47, 48, 52, 50, 53, 50, 53, 52
Offset: 1
Examples
a(12) = 3 since there are 3 ordered ways to write 12 as the sum of two non-refactorable numbers: 5 + 7 = 6 + 6 = 7 + 5.
Programs
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Mathematica
Table[Sum[(Ceiling[k/DivisorSigma[0, k]] - Floor[k/DivisorSigma[0, k]]) (Ceiling[(n - k)/DivisorSigma[0, (n - k)]] - Floor[(n - k)/DivisorSigma[0, (n - k)]]), {k, n - 1}], {n, 100}]
Formula
a(n) = Sum_{k=1..n-1} c(k) * c(n-k), where c = A368625.