A304709 Number of integer partitions of n whose distinct parts are pairwise coprime.

1, 1, 2, 3, 6, 7, 13, 16, 23, 29, 42, 49, 69, 83, 102, 126, 161, 191, 239, 281, 336, 402, 484, 566, 672, 787, 919, 1067, 1251, 1449, 1684, 1934, 2223, 2554, 2920, 3341, 3821, 4344, 4928, 5586, 6334, 7163, 8091, 9100, 10228, 11492, 12902, 14449, 16167, 18058

Offset

1,3

Comments

Two parts are coprime if they have no common divisor greater than 1. For partitions of length 1 note that (1) is coprime but (x) is not coprime for x > 1.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..500

Formula

a(n) = A304712(n) + 1 - A000005(n). - Andrew Howroyd, Nov 02 2019

Example

The a(6) = 7 integer partitions of 6 whose distinct parts are pairwise coprime are (51), (411), (321), (3111), (2211), (21111), (111111).

Mathematica

Table[Select[IntegerPartitions[n], CoprimeQ@@Union[#]&]//Length, {n, 20}]

Prog

(PARI)
lista(nn)={local(Cache=Map());
  my(excl=vector(nn, n, sum(i=1, n-1, if(gcd(i, n)>1, 2^(n-i)))));
  my(c(n, m, b)=
     if(n==0, 1,
        while(m>n || bittest(b, 0), m--; b>>=1);
        my(hk=[n, m, b], z);
        if(!mapisdefined(Cache, hk, &z),
          z = if(m, self()(n, m-1, b>>1) + self()(n-m, m, bitor(b, excl[m])), 0);
          mapput(Cache, hk, z)); z));
  my(a(n)=c(n, n, 0) + 1 - numdiv(n));
  for(n=1, nn, print1(a(n), ", "))
} \\ Andrew Howroyd, Nov 02 2019

Crossrefs

Cf. A000005, A007359, A007360, A018783, A051424, A078374, A101268, A289508, A289509, A302569, A302696, A302698, A302796, A302797, A304711, A304712.

Sequence in context: A018511 A345139 A182708 * A330145 A183558 A328164

Adjacent sequences: A304706 A304707 A304708 * A304710 A304711 A304712

Keyword

nonn

Author

Gus Wiseman, May 17 2018

Status

approved