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A000586 Number of partitions of n into distinct primes.
(Formerly M0022 N0004 N0039)
44
1, 0, 1, 1, 0, 2, 0, 2, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 4, 3, 4, 4, 4, 5, 5, 5, 6, 5, 6, 7, 6, 9, 7, 9, 9, 9, 11, 11, 11, 13, 12, 14, 15, 15, 17, 16, 18, 19, 20, 21, 23, 22, 25, 26, 27, 30, 29, 32, 32, 35, 37, 39, 40, 42, 44, 45, 50, 50, 53, 55, 57, 61, 64, 67, 70, 71, 76, 78, 83, 87, 89, 93, 96 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

REFERENCES

H. Gupta, Certain averages connected with partitions. Res. Bull. Panjab Univ. no. 124 1957 427-430.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence in two entries, N0004 and N0039).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

Edray Herber Goins and Talitha M. Washington, On the generalized climbing stairs problem, Ars Combin.  117  (2014), 183-190.  MR3243840 (Reviewed), arXiv:0909.5459 [math.CO].

H. Gupta, Partitions into distinct primes, Proc. Nat. Acad. Sci. India, 21 (1955), 185-187.

FORMULA

G.f.: Product_{k=1..inf} (1+x^prime(k)).

a(n) = A184171(n) + A184172(n). - R. J. Mathar, Jan 10 2011

a(n) = Sum_{k=0..A024936(n)} A219180(n,k). - Alois P. Heinz, Nov 13 2012

EXAMPLE

n=16 has a(16) = 3 partitions into distinct prime parts: 16 = 2+3+11 = 3+13 = 5+11.

MAPLE

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

      b(n, i-1)+`if`(ithprime(i)>n, 0, b(n-ithprime(i), i-1))))

    end:

a:= n-> b(n, numtheory[pi](n)):

seq(a(n), n=0..100);  # Alois P. Heinz, Nov 15 2012

MATHEMATICA

CoefficientList[Series[Product[(1+x^Prime[k]), {k, 24}], {x, 0, Prime[24]}], x]

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i-1] + If[Prime[i] > n, 0, b[n - Prime[i], i-1]]]]; a[n_] := b[n, PrimePi[n]]; Table[a[n], {n, 0, 100}] (* Jean-Fran├žois Alcover, Apr 09 2014, after Alois P. Heinz *)

PROG

(Haskell)

a000586 = p a000040_list where

   p _  0 = 1

   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m

-- Reinhard Zumkeller, Aug 05 2012

(PARI) a(n, k=n)=if(n<1, !n, my(s); forprime(p=2, k, s+=a(n-p, p-1)); s) \\ Charles R Greathouse IV, Nov 20 2012

CROSSREFS

Cf. A000041, A070215, A000607, A112022, A000607, A000009.

Sequence in context: A191225 A223893 A112022 * A029399 A249338 A046069

Adjacent sequences:  A000583 A000584 A000585 * A000587 A000588 A000589

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane. Entry revised by N. J. A. Sloane, Jun 10 2012

STATUS

approved

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Last modified August 19 01:24 EDT 2017. Contains 290787 sequences.