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.

4365, 74919, 79827, 111507, 347739, 445875, 739557, 2168907, 4481986, 7263945, 7845387, 9309465, 10838247, 12290055, 12673095, 18151479, 22083215, 25645707, 39175955, 62634519, 69076995, 72794967, 80889207, 81166839, 87215967, 94682133, 107522943, 110768835, 119192283

Offset: 1

Author

Claude H. R. Dequatre, Sep 08 2021

Keywords

• nonn
• easy

Comments

Conjecture: the asymptotic density of terms is equal to 0 and this sequence is infinite.

Examples

			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.
		

Links

• Kevin P. Thompson, Table of n, a(n) for n = 1..288

Crossrefs

Cf. A027750, A000005, A000203.

Similar sequences: A002961, A005237, A005238, A006601, A049051, A347076.

Programs

Mathematica

Select[Range[2, 10^6], DivisorSigma[0, #] == 2*DivisorSigma[0, # - 1] && DivisorSigma[1, #] == DivisorSigma[1, # - 1] &] (* Amiram Eldar, Sep 08 2021 *)

PARI

for(k=2,100000000,if(numdiv(k)==2*numdiv(k-1) && sigma(k)==sigma(k-1),print1(k", ")))

Python

from sympy import divisor_count as tau, divisor_sigma as sigma
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