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A338117
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Number of partitions of n into two parts (s,t) such that (t-s) | n, where s < t.
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3
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0, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 3, 3, 1, 2, 1, 3, 3, 1, 1, 5, 2, 1, 3, 3, 1, 3, 1, 4, 3, 1, 3, 5, 1, 1, 3, 5, 1, 3, 1, 3, 5, 1, 1, 7, 2, 2, 3, 3, 1, 3, 3, 5, 3, 1, 1, 7, 1, 1, 5, 5, 3, 3, 1, 3, 3, 3, 1, 8, 1, 1, 5, 3, 3, 3, 1, 7, 4, 1, 1, 7, 3, 1, 3, 5, 1, 5, 3, 3, 3
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OFFSET
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1,8
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COMMENTS
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The above observation is true, which can be seen from the formula A320111(2n) = A000005(n), A320111(2n+1) = A000005(2n+1). For odd numbers, the difference (t-s) may range over all the divisors of n except the n itself, and for even numbers the difference (t-s) [which is always even] may range only over the even divisors of n, except the n itself. Note that A000005(2n) = A000005(n) + A001227(n). - Antti Karttunen, Dec 12 2021
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LINKS
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FORMULA
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a(n) = Sum_{i=1..floor((n-1)/2)} (1 - ceiling(n/(n-2*i)) + floor(n/(n-2*i))).
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EXAMPLE
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a(8) = 2; The partitions of 8 into two parts (s,t) such that s < t are (7,1), (6,2), (5,3) and (4,4). Only the partitions (6,2) and (5,3) have (6-2) | 8 and (5-3) | 8, so a(8) = 2.
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MATHEMATICA
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Table[Sum[(1 - Ceiling[n/(n - 2 i)] + Floor[n/(n - 2 i)]), {i, Floor[(n - 1)/2]}], {n, 100}]
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PROG
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(PARI) for(n=1, 85, my(j=0); forpart(x=n, if(#x==2, if(x[2]!=x[1]&&!(n%(x[2]-x[1])), j++))); print1(j, ", ")) \\ Hugo Pfoertner, Oct 30 2020
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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