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

1,1

Comments

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

Links

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

Cf. A027750, A000005, A000203.

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

Sequence in context: A125825 A031564 A359490 * A217345 A140937 A327789

Adjacent sequences: A347600 A347601 A347602 * A347604 A347605 A347606

Keyword

nonn,easy

Author

Claude H. R. Dequatre, Sep 08 2021

Status

approved