0,1

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

"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 enen among the first N values of p(n) for any givan N are known (Scott Ahlgren; and Nicolas, Rusza and Sarkozy 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."

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)

H. Gupta, A note on the parity of p(n), J. Indian Math. Soc. (N.S.) 10, (1946). 32-33. MR0020588 (8,566g)

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)

Hirschhorn, M. D. On the parity of p(n), II, J. Combin. Theory Ser. A 62 (1993), no. 1, 128-138.

Hirschhorn, M. D. and Subbarao, M. V. 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.

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. 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)

M. V. Subbarao, A note on the parity of p(n), Indian J. Math. 14 (1972), 147-148. MR0357355 (50 #9823)

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.

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

a(n) = pp(n, 1), with Boolean pp(n, k) = if k<n then pp(n-k, k) XOR pp(n, k+1) else (k=n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 04 2003

a(n) = Pm(n,1) with Pm(n,k) = if k<n then (Pm(n-k,k) + Pm(n,k+1)) mod 2 else 0^(n*(k-n)). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 09 2009]

(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)

Cf. A000041, A071640, A086144.

Sequence in context: A105368 A138019 A097343 this_sequence A108788 A103583 A070178

Adjacent sequences: A040048 A040049 A040050 this_sequence A040052 A040053 A040054

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).