A378794 - OEIS

%I #11 Dec 21 2024 00:45:17

%S 1,16,36,100,196,225,256,441,484,625,676,1089,1156,1225,1296,2116,

%T 2601,3136,3249,3364,3844,4225,5929,6561,7056,7225,7569,8100,8281,

%U 8836,9216,10000,11236,12321,13924,14161,14884,15129,16641,17689,17956,19881,20164,20449

%N Numbers k such that tau(k) == 1 (mod(2*(tau(k^2 + k - 1)))), where tau(k) = A000005(k).

%C 736 terms < 10^7 were found.

%C All terms are squares because their number of divisors is odd (see formula field in A000005: a(n) is odd iff n is square).

%e 1 is a term because tau(1) = 1, tau(1 + 1 - 1) = 1 and 1 modulo 2 is 1.

%e 16 is a term because tau(16) = 5, tau(256 + 16 - 1) = 2 and 5 modulo 4 is 1.

%e 50 is not a term because tau(50) = 6, tau(2500 + 50 -1) = 2 and 6 modulo 4 is 2.

%o (PARI) isok(k)=my(d_1=numdiv(k),d_2=numdiv(k^2+k-1));d_1%(2*d_2)==1;

%o for(k=1,1000,if(isok(k),print1(k", ")))

%Y Cf. A000005, A000040.

%Y Cf. A378789, A378792.

%K nonn,easy

%O 1,2

%A _Claude H. R. Dequatre_, Dec 07 2024