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A355058 Numbers m such that d(m) mod 6 = 3, where d(m) is the number of divisors of m.

%I A355058 #18 Jun 16 2025 23:49:23
%S A355058 4,9,25,36,49,100,121,144,169,196,225,256,289,324,361,400,441,484,529,
%T A355058 576,676,784,841,900,961,1089,1156,1225,1369,1444,1521,1600,1681,1764,
%U A355058 1849,1936,2025,2116,2209,2304,2500,2601,2704,2809,2916,3025,3136,3249,3364
%N A355058 Numbers m such that d(m) mod 6 = 3, where d(m) is the number of divisors of m.
%C A355058 All terms are square; contains squares of primes.
%H A355058 Amiram Eldar, <a href="/A355058/b355058.txt">Table of n, a(n) for n = 1..10000</a>
%F A355058 { A000290 \ A352475 }.
%F A355058 Sum_{n>=1} 1/a(n) = Pi^2/18 (A086463). - _Amiram Eldar_, Jul 06 2022
%e A355058 Let p be a prime; p^2 has 3 divisors {1, p, p^2}, therefore all squares of primes {4, 9, 25, 49, ...} are in the sequence.
%e A355058 36 is in the sequence because d(36) = 9, and 9 mod 6 = 3.
%e A355058 16 is not in the sequence because it has 5 divisors, and 5 mod 6 = 5.
%t A355058 Select[Range[2^12], Mod[DivisorSigma[0, #], 6] == 3 &]
%o A355058 (PARI) isok(m) = (numdiv(m) % 6) == 3; \\ _Michel Marcus_, Jul 05 2022
%o A355058 (Python)
%o A355058 from itertools import count, islice
%o A355058 from sympy import factorint, prod
%o A355058 def A355058_gen(): # generator of terms
%o A355058     return map(lambda n:n**2,filter(lambda n:prod((2*e+1)%6 for e in factorint(n).values())%6==3,count(1)))
%o A355058 A355058_list = list(islice(A355058_gen(),30)) # _Chai Wah Wu_, Jul 06 2022
%Y A355058 Cf. A000005, A000290, A001248, A086463, A352475.
%K A355058 nonn,easy
%O A355058 1,1
%A A355058 _Michael De Vlieger_, Jul 04 2022