A057562 Number of partitions of n into parts all relatively prime to n.

1, 1, 2, 2, 6, 2, 14, 6, 16, 7, 55, 6, 100, 17, 44, 32, 296, 14, 489, 35, 178, 77, 1254, 30, 1156, 147, 731, 142, 4564, 25, 6841, 390, 1668, 474, 4780, 114, 21636, 810, 4362, 432, 44582, 103, 63260, 1357, 4186, 2200, 124753, 364, 105604, 1232, 24482, 3583

Offset: 1

Author

Leroy Quet, Oct 03 2000

Keywords

• nonn

Comments

p is prime iff a(p) = A000041(p)-1. - Lior Manor, Feb 04 2005

Examples

			The unrestricted partitions of 4 are 1+1+1+1, 1+1+2, 1+3, 2+2 and 4. Of these, only 1+1+1+1 and 1+3 contain parts which are all relatively prime to 4. So a(4) = 2.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A036998, A038566, A100347, A227296.

See also A098743 (parts don't divide n).

Programs

Haskell

a057562 n = p (a038566_row n) n where
   p _          0 = 1
   p []         _ = 0
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Jul 05 2013

Mathematica

Table[Count[IntegerPartitions@ n, k_ /; AllTrue[k, CoprimeQ[#, n] &]], {n, 52}] (* Michael De Vlieger, Aug 01 2017 *)

PARI

R(n, v)=if(#v<2 || n=0, sum(i=1, #v, R(n-v[i], v[1..i])))
a(n)=if(isprime(n), return(numbpart(n)-1)); R(n, select(k->gcd(k, n)==1, vector(n, i, i))) \\ Charles R Greathouse IV, Sep 13 2012

PARI

a(n)=polcoeff(1/prod(k=1,n,if(gcd(k,n)==1,1-x^k,1), O(x^(n+1))+1), n) \\ Charles R Greathouse IV, Sep 13 2012

Formula

Coefficient of x^n in expansion of 1/Product_{d : gcd(d, n)=1} (1-x^d). - Vladeta Jovovic, Dec 23 2004

Extensions

More terms from Naohiro Nomoto, Feb 28 2002

Corrected by Vladeta Jovovic, Dec 23 2004