

A057562


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


12



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
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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} (1x^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(nv[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, 1x^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



