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 A000837 Number of partitions of n into relatively prime parts. Also aperiodic partitions. 174
 1, 1, 1, 2, 3, 6, 7, 14, 17, 27, 34, 55, 63, 100, 119, 167, 209, 296, 347, 489, 582, 775, 945, 1254, 1481, 1951, 2334, 2980, 3580, 4564, 5386, 6841, 8118, 10085, 12012, 14862, 17526, 21636, 25524, 31082, 36694, 44582, 52255, 63260, 74170, 88931, 104302 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Starting (1, 1, 2, 3, 6, 7, 14, ...), = row sums of triangle A137585. - Gary W. Adamson, Jan 27 2008 Triangle A168532 has aerated variants of this sequence in each column starting with offset 1, row sums = A000041. -  Gary W. Adamson, Nov 28 2009 A partition is aperiodic iff its multiplicities are relatively prime, i.e., its Heinz number (A215366) is not a perfect power (A007916). - Gus Wiseman, Dec 19 2017 REFERENCES H. W. Gould, personal communication. LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe) Mohamed El Bachraoui, On the Parity of p(n,3) and p_psi(n,3), Contributions to Discrete Mathematics, Vol. 5.10 (2010). Mircea Merca and Maxie D. Schmidt, Generating Special Arithmetic Functions by Lambert Series Factorizations, arXiv:1706.00393 [math.NT], 2017. See Remark 3.4. N. J. A. Sloane, Transforms FORMULA Möbius transform of A000041. - Christian G. Bower, Jun 11 2000 Product_{n>0} 1/(1-q^n) = 1 + Sum_{n>0} a(n)*q^n/(1-q^n). - Mamuka Jibladze, Nov 14 2015 a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*n*sqrt(3)). - Vaclav Kotesovec, Jan 28 2019 EXAMPLE Of the 11 partitions of 6, we must exclude 6, 4+2, 3+3 and 2+2+2, so a(6) = 11 - 4 = 7. For n=6, 2+2+1+1 is periodic because it can be written 2*(2+1), similarly 1+1+1+1+1+1, 3+3 and 2+2+2. The a(6) = 7 partitions into relatively prime parts are (51), (411), (321), (3111), (2211), (21111), (111111). The a(6) = 7 aperiodic partitions are (6), (51), (42), (411), (321), (3111), (21111). - Gus Wiseman, Dec 19 2017 MATHEMATICA p[n_] := IntegerPartitions[n]; l[n_] := Length[p[n]]; g[n_, j_] := Apply[GCD, Part[p[n], j]] h[n_] := Table[g[n, j], {j, 1, l[n]}] Join[{0}, Table[Count[h[n], 1], {n, 1, 20}]] (* Clark Kimberling, Mar 09 2012 *) a = 1; a[n_] := Sum[ MoebiusMu[n/d] * PartitionsP[d], {d, Divisors[n]}]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Oct 03 2013 *) PROG (PARI) N=66; x='x+O('x^N); gf=2+sum(n=1, N, (1/eta(x^n))*moebius(n)); Vec(gf) \\ Joerg Arndt, May 11 2013 (Python) from sympy import npartitions, mobius, divisors def a(n): return 1 if n==0 else sum([mobius(n/d)*npartitions(d) for d in divisors(n)]) # Indranil Ghosh, Apr 26 2017 CROSSREFS Cf. A000740, A007916, A047968, A055892, A100953, A137585, A168532, A281116. Sequence in context: A117087 A322367 A319811 * A200144 A056498 A325093 Adjacent sequences:  A000834 A000835 A000836 * A000838 A000839 A000840 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS Corrected and extended by David W. Wilson, Aug 15 1996 Additional name from Christian G. Bower, Jun 11 2000 STATUS approved

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Last modified September 20 10:10 EDT 2019. Contains 327229 sequences. (Running on oeis4.)