A040051 Parity of partition function A000041.

1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

From M. V. Subbarao (m.v.subbarao(AT)ualberta.ca), Sep 05 2003: (Start)

Essentially this same question was raised by Ramanujan in a letter to P. A. MacMahon around 1920 (see page 1087, MacMahon's Collected Papers). With the help of Jacobi's triple product identity, MacMahon showed that p(1000) is odd (as he says, with five minutes work - there were no computers those days).

Now we know that among the first ten million values of p(n) 5002137 of them are odd. It is conjectured (T. R. Parkin and D. Shanks) that p(n) is equally often even and odd. Lower bound estimates for the number of times p(n) is even among the first N values of p(n) for any given N are known (Scott Ahlgren; and Nicolas, Rusza and Sárközy among others).

Earlier this year a remarkable result was proved by Boylan and Ahlgren (AMS ABSTRACT # 987-11-82) which says that beyond the three eighty-year old Ramanujan congruences - namely, p(5n+4), p(7n+5) and p(11n +6) being divisible respectively by 5,7 and 11 - there are no other simple congruences of this kind.

My 1966 conjecture that in every arithmetic progression r (mod s) for arbitrary integral r and s, there are infinitely many integers n for which p(n) is odd - with a similar statement for p(n) even - was proved for the even case by Ken Ono (1996) and for the odd case for all s up to 10^5 and for all s which are powers of 2 by Bolyan and Ono, 2002.

(End)

a(n) is also the parity of the trace Tr(n) = A183011(n), the numerator of the Bruinier-Ono formula for the partition function, if n >= 1. - Omar E. Pol, Mar 14 2012

Consider the diagram of the regions of n (see A206437). Then, in each odd-indexed region of n, fill each part of size k with k 1's. Then, in each even-indexed region of n, fill each part of size k with k 0's. The successive digits of row 1 of the diagram give the first n elements of this sequence, if n >= 1. - Omar E. Pol, May 02 2012

References

• H. Gupta, A note on the parity of p(n), J. Indian Math. Soc. (N.S.) 10, (1946). 32-33. MR0020588 (8,566g)
• K. M. Majumdar, On the parity of the partition function p(n), J. Indian Math. Soc. (N.S.) 13, (1949). 23-24. MR0030553 (11,13d)
• M. V. Subbarao, A note on the parity of p(n), Indian J. Math. 14 (1972), 147-148. MR0357355 (50 #9823)

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• R. Blecksmith; J. Brillhart; I. Gerst, Parity results for certain partition functions and identities similar to theta function identities, Math. Comp. 48 (1987), no. 177, 29-38. MR0866096 (87k:11113).
• Nicholas Eriksson, q-series, elliptic curves and odd values of the partition function, Int. J. Math. Math. Sci. 22 (1999), 55-65; MR 2001a:11175.
• M. D. Hirschhorn, On the residue mod 2 and mod 4 of p(n), Acta Arith. 38 (1980/81), no. 2, 105-109. MR0604226 (82d:10025)
• M. D. Hirschhorn, On the parity of p(n), II, J. Combin. Theory Ser. A 62 (1993), no. 1, 128-138.
• M. D. Hirschhorn and M. V. Subbarao, On the parity of p(n), Acta Arith. 50 (1988), no. 4, 355-356.
• O. Kolberg, Note on the parity of the partition function, Math. Scand. 7 1959 377-378. MR0117213 (22 #7995).
• P. A. MacMahon, The parity of p(n), the number of partitions of n, when n <= 1000, J. London Math. Soc., 1 (1926), 225-226.
• Mircea Merca, New recurrences for Euler's partition function, Turkish J. Math. 41:5 (2017), pp. 1184-1190.
• M. Newman, Periodicity modulo m and divisibility properties of the partition function, Trans. Amer. Math. Soc. 97 (1960), 225-236. MR0115981 (22 #6778)
• M. Newman, Congruences for the partition function to composite moduli, Illinois J. Math. 6 1962 59-63. MR0140472 (25 #3892)
• K. Ono, Parity of the partition function, Electron. Res. Announc. AMS, Vol. 1, 1995, pp. 35-42; MR 96d:11108.
• Ivars Peterson, Ken Ono's and Nicholas Eriksson's work

Crossrefs

Cf. A000041, A071640, A086144, A190938.

Programs

Haskell

import Data.Bits (xor)
a040051 n = p 1 n :: Int where
   p _ 0 = 1
   p k m | k <= m = p k (m - k) `xor` p (k+1) m | k > m  = 0
-- Reinhard Zumkeller, Nov 15 2011

Mathematica

Table[ Mod[ PartitionsP@ n, 2], {n, 105}] (* Robert G. Wilson v, Mar 25 2011 *)

PARI

a(n)=if(n<0, 0, numbpart(n)%2)

PARI

a(n)=if(n<0, 0, polcoeff(1/eta(x+x*O(x^n)), n)%2)

PARI

a(n)=if(n<10^9, return(numbpart(n)%2)); my(r=n%4, u=select(k->k^2%32==8*r+1,[1..31]), st=u[1], m=n\4, s); u=[u[2]-u[1],u[3]-u[2],u[4]-u[3],u[1]+32-u[4]]; forstep(t=[1,3,7,5][r+1], sqrtint(32*m-1), u, k=t^2>>5; if(a(m-k), s++)); s%2 \\ Merca's algorithm, switching to direct computation for n less than 10^9. Very time-consuming but low memory use. - Charles R Greathouse IV, Jan 24 2018

Python

from sympy import npartitions
def a(n): return npartitions(n)%2 # Indranil Ghosh, May 25 2017

Formula

a(n) = pp(n, 1), with Boolean pp(n, k) = if kReinhard Zumkeller, Sep 04 2003

a(n) = Pm(n,1) with Pm(n,k) = if kReinhard Zumkeller, Jun 09 2009

a(n) = A000035(A000041(n)). - Omar E. Pol, Aug 05 2013

a(n) = A000035(A000025(n)). - John M. Campbell, Jun 29 2016