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

1,3

Comments

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

Links

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

Formula

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

Example

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.

Mathematica

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

Prog

(PARI) R(n, v)=if(#v<2 || n<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
(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

Crossrefs

Cf. A036998, A038566, A100347, A227296.

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

Sequence in context: A096217 A281145 A098555 * A102628 A211776 A325248

Adjacent sequences: A057559 A057560 A057561 * A057563 A057564 A057565

Keyword

nonn

Author

Leroy Quet, Oct 03 2000

Extensions

More terms from Naohiro Nomoto, Feb 28 2002

Corrected by Vladeta Jovovic, Dec 23 2004

Status

approved