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A304709 Number of integer partitions of n whose distinct parts are pairwise coprime. 6
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 (list; graph; refs; listen; history; text; internal format)
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: A167415 A018511 A182708 * A183558 A328164 A294916

Adjacent sequences:  A304706 A304707 A304708 * A304710 A304711 A304712

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 17 2018

STATUS

approved

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Last modified December 13 03:41 EST 2019. Contains 329968 sequences. (Running on oeis4.)