A065061 Numbers k such that sigma(k) - tau(k) is a prime.

3, 8, 162, 512, 1250, 8192, 31250, 32768, 41472, 663552, 2531250, 3748322, 5120000, 6837602, 7558272, 8000000, 15780962, 33554432, 35701250, 42762752, 45334242, 68024448, 75031250, 78125000, 91125000, 137149922, 243101250, 512000000, 907039232, 959570432

Offset: 1

Author

Jason Earls, Nov 06 2001

Keywords

• nonn

Comments

From Kevin P. Thompson, Jun 20 2022: (Start)

Terms greater than 3 must be twice a square (see A064205).

No terms are congruent to 4 or 6 (mod 10) (see A064205).

(End)

Examples

			162 is a term since sigma(162) - tau(162) = 363 - 10 = 353, which is prime.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..5000 (terms 1..265 from Kevin P. Thompson)

Crossrefs

I.e., A065608(n) is prime. Cf. A064205.

Cf. A000005, A000203, A023194, A038344, A055813, A062700, A115919, A141242, A229264, A229265, A229266, A229268.

Programs

Mathematica

Do[ If[ PrimeQ[ DivisorSigma[1, n] - DivisorSigma[0, n]], Print[n]], {n, 1, 10^7}]

PARI

{ n=0; for (m=1, 10^9, if (isprime(sigma(m) - numdiv(m)), write("b065061.txt", n++, " ", m); if (n==100, return)) ) } \\ Harry J. Smith, Oct 05 2009

Python

from itertools import count, islice
from sympy import isprime, divisor_sigma as s, divisor_count as t
def agen(): # generator of terms
    yield 3
    yield from (k for k in (2*i*i for i in count(1)) if isprime(s(k)-t(k)))
print(list(islice(agen(), 30))) # Michael S. Branicky, Jun 20 2022

Extensions

a(17)-a(28) from Harry J. Smith, Oct 05 2009

a(29)-a(30) from Kevin P. Thompson, Jun 20 2022