

A040051


Parity of partition function A000041.


25



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
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OFFSET

0,1


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 # 9871182) which says that beyond the three eightyyear 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 BruinierOno 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 oddindexed region of n, fill each part of size k with k 1's. Then, in each evenindexed 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). 3233. MR0020588 (8,566g)
K. M. Majumdar, On the parity of the partition function p(n), J. Indian Math. Soc. (N.S.) 13, (1949). 2324. MR0030553 (11,13d)
M. V. Subbarao, A note on the parity of p(n), Indian J. Math. 14 (1972), 147148. 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, 2938. MR0866096 (87k:11113).
Nicholas Eriksson, qseries, elliptic curves and odd values of the partition function, Int. J. Math. Math. Sci. 22 (1999), 5565; MR 2001a:11175.
M. D. Hirschhorn, On the residue mod 2 and mod 4 of p(n), Acta Arith. 38 (1980/81), no. 2, 105109. MR0604226 (82d:10025)
M. D. Hirschhorn, On the parity of p(n), II, J. Combin. Theory Ser. A 62 (1993), no. 1, 128138.
M. D. Hirschhorn and M. V. Subbarao, On the parity of p(n), Acta Arith. 50 (1988), no. 4, 355356.
O. Kolberg, Note on the parity of the partition function, Math. Scand. 7 1959 377378. 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), 225226.
Mircea Merca, New recurrences for Euler's partition function, Turkish J. Math. 41:5 (2017), pp. 11841190.
M. Newman, Periodicity modulo m and divisibility properties of the partition function, Trans. Amer. Math. Soc. 97 (1960), 225236. MR0115981 (22 #6778)
M. Newman, Congruences for the partition function to composite moduli, Illinois J. Math. 6 1962 5963. MR0140472 (25 #3892)
K. Ono, Parity of the partition function, Electron. Res. Announc. AMS, Vol. 1, 1995, pp. 3542; MR 96d:11108.
Ivars Peterson, Ken Ono's and Nicholas Eriksson's work


FORMULA

a(n) = pp(n, 1), with Boolean pp(n, k) = if k<n then pp(nk, k) XOR pp(n, k+1) else (k=n).  Reinhard Zumkeller, Sep 04 2003
a(n) = Pm(n,1) with Pm(n,k) = if k<n then (Pm(nk,k) + Pm(n,k+1)) mod 2 else 0^(n*(kn)).  Reinhard 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


MATHEMATICA

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


PROG

(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]+32u[4]]; forstep(t=[1, 3, 7, 5][r+1], sqrtint(32*m1), u, k=t^2>>5; if(a(mk), s++)); s%2 \\ Merca's algorithm, switching to direct computation for n less than 10^9. Very timeconsuming but low memory use.  Charles R Greathouse IV, Jan 24 2018
(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
(Python)
from sympy import npartitions
def a(n): return npartitions(n)%2 # Indranil Ghosh, May 25 2017


CROSSREFS

Cf. A000041, A071640, A086144, A190938.
Sequence in context: A267919 A097343 A188011 * A108788 A103583 A070178
Adjacent sequences: A040048 A040049 A040050 * A040052 A040053 A040054


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane


STATUS

approved



