A064205 - OEIS

%I #50 Dec 06 2022 07:56:51

%S 1,2,8,128,162,512,32768,41472,101250,125000,1414562,3748322,5120000,

%T 6837602,8000000,13530402,24234722,35701250,66724352,75031250,

%U 78125000,86093442,91125000,171532242,177058562,226759808,233971712,617831552,664301250,686128968

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

%C The terms involve powers of small primes. - _Jud McCranie_, Nov 29 2001

%C From _Kevin P. Thompson_, Jun 20 2022: (Start)

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

%C 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.

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

%C 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)

%H Amiram Eldar, <a href="/A064205/b064205.txt">Table of n, a(n) for n = 1..5000</a> (first 34 terms from Harry J. Smith, terms 35..276 from Kevin P. Thompson)

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

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

%o (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

%o (Python)

%o from itertools import count, islice

%o from sympy import isprime, divisor_sigma as s, divisor_count as t

%o def agen(): # generator of terms

%o     yield 1

%o     yield from (k for k in (2*i*i for i in count(1)) if isprime(s(k)+t(k)))

%o print(list(islice(agen(), 30))) # _Michael S. Branicky_, Jun 20 2022

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

%K nonn

%O 1,2

%A _Jason Earls_, Sep 21 2001

%E More terms from _Robert G. Wilson v_, Nov 12 2001

%E More terms from _Labos Elemer_, Nov 22 2001

%E More terms from _Jud McCranie_, Nov 29 2001

%E a(28) from _Harry J. Smith_, Sep 10 2009