A064205 Numbers k such that sigma(k) + tau(k) is a prime.

1, 2, 8, 128, 162, 512, 32768, 41472, 101250, 125000, 1414562, 3748322, 5120000, 6837602, 8000000, 13530402, 24234722, 35701250, 66724352, 75031250, 78125000, 86093442, 91125000, 171532242, 177058562, 226759808, 233971712, 617831552, 664301250, 686128968

Offset: 1

Author

Jason Earls, Sep 21 2001

Keywords

• nonn

Comments

The terms involve powers of small primes. - Jud McCranie, Nov 29 2001

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

Theorem: Terms that are greater than one must be twice a square.

Proof: Since sigma(k) is odd if and only if k is a square or twice a square, and tau(k) is odd if and only if k is a square, then an odd sum only occurs when k is twice a square, in which case sigma(k) is odd and tau(k) is even. So, these are the only candidates for sigma(k) + tau(k) being prime.

Theorem: No terms are congruent to 4 or 6 (mod 10).

Proof: Since no square ends in 2, 3, 7, or 8, and each term > 1 is twice a square, no term ends in 4 or 6. (End)

Examples

			128 is a term since sigma(128) + tau(128) = 255 + 8 = 263, which is prime.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..5000 (first 34 terms from Harry J. Smith, terms 35..276 from Kevin P. Thompson)

Crossrefs

Cf. A007503 (sigma+tau), A065061, A055813.

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("b064205.txt", n++, " ", m); if (n==100, break)) ) } \\ Harry J. Smith, Sep 10 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 1
    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

More terms from Robert G. Wilson v, Nov 12 2001

More terms from Labos Elemer, Nov 22 2001

More terms from Jud McCranie, Nov 29 2001

a(28) from Harry J. Smith, Sep 10 2009