The partition function modulo 3 in arithmetic progressions

Let be the partition function. Ahlgren and Ono conjectured that every arithmetic progression contains infinitely many integers for which is not congruent to . Radu proved this conjecture in 2010 using the work of Deligne and Rapoport. In this note, we give a simpler proof of Ahlgren and Ono's conjecture in the special case where the modulus of the arithmetic progression is a power of by applying a method of Boylan and Ono and using the work of Bella < che and Khare generalizing Nicolas and Serre's results on the local nilpotency of the Hecke algebra.