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 A047966 a(n) = Sum_{ d divides n } q(d), where q(d) = A000009 = number of partitions of d into distinct parts. 81
 1, 2, 3, 4, 4, 8, 6, 10, 11, 15, 13, 25, 19, 29, 33, 42, 39, 62, 55, 81, 84, 103, 105, 153, 146, 185, 203, 253, 257, 344, 341, 432, 463, 552, 594, 747, 761, 920, 1003, 1200, 1261, 1537, 1611, 1921, 2089, 2410, 2591, 3095, 3270, 3815, 4138, 4769, 5121, 5972, 6394, 7367, 7974, 9066, 9793, 11305, 12077, 13736, 14940 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Number of partitions of n such that every part occurs with the same multiplicity. - Vladeta Jovovic, Oct 22 2004 Christopher and Christober call such partitions uniform. - Gus Wiseman, Apr 16 2018 Equals inverse Mobius transform (A051731) * A000009, where the latter begins (1, 1, 2, 2, 3, 4, 5, 6, 8, ...). -  Gary W. Adamson, Jun 08 2009 LINKS T. D. Noe and Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe) A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp.1-12. FORMULA G.f.: Sum_{k>0} (-1+Product_{i>0} (1+z^(k*i))). - Vladeta Jovovic, Jun 22 2003 G.f.: Sum_{k>=1} q(k)*x^k/(1 - x^k), where q() = A000009. - Ilya Gutkovskiy, Jun 20 2018 a(n) ~ exp(Pi*sqrt(n/3)) / (4*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Aug 27 2018 EXAMPLE The a(6) = 8 uniform partitions are (6), (51), (42), (33), (321), (222), (2211), (111111). - Gus Wiseman, Apr 16 2018 MAPLE with(numtheory): b:= proc(n) option remember; `if`(n=0, 1, add(add(      `if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)     end: a:= n-> add(b(d), d=divisors(n)): seq(a(n), n=1..100);  # Alois P. Heinz, Jul 11 2016 MATHEMATICA b[n_] := b[n] = If[n==0, 1, Sum[DivisorSum[j, If[OddQ[#], #, 0]&]*b[n-j], {j, 1, n}]/n]; a[n_] := DivisorSum[n, b]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 06 2016 after Alois P. Heinz *) Table[DivisorSum[n, PartitionsQ], {n, 20}] (* Gus Wiseman, Apr 16 2018 *) PROG (PARI) N = 66; q='q+O('q^N); D(q)=eta(q^2)/eta(q); \\ A000009 Vec( sum(e=1, N, D(q^e)-1) ) \\ Joerg Arndt, Mar 27 2014 CROSSREFS Cf. A000009, A000837, A024994, A047968, A063834, A279788, A289501, A300383, A301462, A302698. Sequence in context: A236129 A240219 A028298 * A317085 A236543 A319079 Adjacent sequences:  A047963 A047964 A047965 * A047967 A047968 A047969 KEYWORD nonn AUTHOR STATUS approved

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Last modified May 7 20:36 EDT 2021. Contains 343652 sequences. (Running on oeis4.)