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 A269926 Number of partitions of n into rational parts a/b such that 1<=a,b<=n and gcd(a,b)=1. 0
 1, 4, 33, 385, 11483, 305684 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS A018805 is the number of rational parts a/b, such that 1<=a,b<=n and gcd(a,b)=1. LINKS EXAMPLE For n=2, the rational parts a/b, such that 1<=a,b<= n and gcd(a,b)=1, are: { 1/1, 1/2, 2/1 }. a(2)=4 because 2 can be partitioned into the following 4 partitions: { 1/2, 1/2, 1/2, 1/2 }, { 1/1, 1/2, 1/2 }, { 1/1, 1/1 }, { 2/1 }. PROG (Sage) from itertools import combinations_with_replacement seq = [] for n in range( 1, 5 ): ....rationals = set() ....for a in range( 1, n+1 ): ........for b in range( 1, n+1 ): ............rational = Rational( (a, b) ) ............rationals.add( rational ) ....partition_count = 0 ....for r in range( 1, n^2 + 1 ): ........for partition in combinations_with_replacement( rationals, r ): ............if sum( partition ) == n: ................partition_count += 1 ....seq.append( partition_count ) print seq (Sage)# Faster version def count_combinations( n, values, r ): ....combo = [ None ] * r ....level = 0 ....min_index = 0 ....count = 0 ....return get_count( n, values, r, combo, level, min_index, count ) def get_count( n, values, r, combo, level, min_index, count ): ....if level < r: ........for i in range( min_index, len( values ) ): ............combo[level] = values[i] ............if sum( combo[0:level] ) < n: ................count = get_count( n, values, r, combo, level+1, i, count ) ....else: ........if sum( combo ) == n: ............count += 1 ....return count seq = [] for n in range( 1, 5 ): ....rational_set = set() ....for a in range( 1, n+1 ): ........for b in range( 1, n+1 ): ............rational = Rational( (a, b) ) ............rational_set.add( rational ) ....rationals = sorted( list( rational_set ) ) ....partition_count = 0 ....for r in range( 1, n^2 + 1 ): ........partition_count += count_combinations( n, rationals, r ) ....seq.append( partition_count ) print seq CROSSREFS Cf. A018805, A209489. Sequence in context: A293020 A293193 A295256 * A156132 A215364 A213641 Adjacent sequences:  A269923 A269924 A269925 * A269927 A269928 A269929 KEYWORD nonn,more AUTHOR Robert C. Lyons, Mar 07 2016 STATUS approved

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Last modified January 19 06:37 EST 2020. Contains 331033 sequences. (Running on oeis4.)