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A347603 Numbers k such that tau(k) = 2*tau(k-1) and sigma(k) = sigma(k-1), where tau(k) and sigma(k) are respectively the number and sum functions of the divisors of k.

%I A347603 #24 Jan 17 2022 11:27:32
%S A347603 4365,74919,79827,111507,347739,445875,739557,2168907,4481986,7263945,
%T A347603 7845387,9309465,10838247,12290055,12673095,18151479,22083215,
%U A347603 25645707,39175955,62634519,69076995,72794967,80889207,81166839,87215967,94682133,107522943,110768835,119192283
%N A347603 Numbers k such that tau(k) = 2*tau(k-1) and sigma(k) = sigma(k-1), where tau(k) and sigma(k) are respectively the number and sum functions of the divisors of k.
%C A347603 Conjecture: the asymptotic density of terms is equal to 0 and this sequence is infinite.
%H A347603 Kevin P. Thompson, <a href="/A347603/b347603.txt">Table of n, a(n) for n = 1..288</a>
%e A347603 a(1) = 4365 because the divisors of 4365 are: 1, 3, 5, 9, 15, 45, 97, 291, 485, 873, 1455, 4365; so, tau(4365) = 12 and sigma(4365) = 7644. The divisors of 4364 are: 1, 2, 4, 1091, 2182, 4364; so, tau(4364) = 6 and sigma(4364) = 7644. Thus tau(4365) = 2*tau(4364), sigma(4365) = sigma(4364) and so 4365 is a term.
%t A347603 Select[Range[2, 10^6], DivisorSigma[0, #] == 2*DivisorSigma[0, # - 1] && DivisorSigma[1, #] == DivisorSigma[1, # - 1] &] (* _Amiram Eldar_, Sep 08 2021 *)
%o A347603 (PARI) for(k=2,100000000,if(numdiv(k)==2*numdiv(k-1) && sigma(k)==sigma(k-1),print1(k", ")))
%o A347603 (Python) from sympy import divisor_count as tau, divisor_sigma as sigma
%o A347603 print([k for k in range(2, 10**6) if tau(k) == 2*tau(k-1) and sigma(k) == sigma(k-1)]) # _Karl-Heinz Hofmann_, Jan 15 2022
%Y A347603 Cf. A027750, A000005, A000203.
%Y A347603 Similar sequences: A002961, A005237, A005238, A006601, A049051, A347076.
%K A347603 nonn,easy
%O A347603 1,1
%A A347603 _Claude H. R. Dequatre_, Sep 08 2021