▁Ian ▁G . ▁Mac don ald ▁was ▁given ▁in ▁. ▁ ▁Val ues ▁The ▁first ▁few ▁values ▁of ▁the ▁t au ▁function ▁are ▁given ▁in ▁the ▁following ▁table ▁: ▁ ▁Ram an uj an ' s ▁conject ures ▁ ▁observed , ▁but ▁did ▁not ▁prove , ▁the ▁following ▁three ▁properties ▁of ▁: ▁▁▁ ▁if ▁ ▁( mean ing ▁that ▁ ▁is ▁a ▁multip lic ative ▁function ) ▁▁ ▁for ▁p ▁prime ▁and ▁r   >   0 . ▁▁ ▁for ▁all ▁pr imes ▁p . ▁ ▁The ▁first ▁two ▁properties ▁were ▁proved ▁by ▁ ▁and ▁the ▁third ▁one , ▁called ▁the ▁Ram an uj an ▁conject ure , ▁was ▁proved ▁by ▁Del igne ▁in ▁ 1 9 7 4 ▁as ▁a ▁consequence ▁of ▁his ▁proof ▁of ▁the ▁We il ▁conject ures ▁( specific ally , ▁he ▁dedu ced ▁it ▁by ▁applying ▁them ▁to ▁a ▁K uga - S ato ▁variety ). ▁ ▁Cong ru ences ▁for ▁the ▁t au ▁function ▁For ▁k   ∈   Z ▁and ▁n   ∈   Z > 0 , ▁define ▁ σ k ( n ) ▁as ▁the ▁sum ▁of ▁the ▁k - th ▁powers ▁of ▁the ▁divis ors ▁of ▁n . ▁The ▁t au ▁function ▁satisfies ▁several ▁con gru ence ▁relations ; ▁many ▁of ▁them ▁can ▁be ▁expressed ▁in ▁terms ▁of ▁ σ k ( n ). ▁Here ▁are ▁some : ▁ ▁For ▁p   ≠   2 3 ▁prime , ▁we ▁have ▁< li > ▁< li > ▁< li > ▁ ▁Con ject ures