cp's OEIS Frontend

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

%I A064205 #50 Dec 06 2022 07:56:51
%S A064205 1,2,8,128,162,512,32768,41472,101250,125000,1414562,3748322,5120000,
%T A064205 6837602,8000000,13530402,24234722,35701250,66724352,75031250,
%U A064205 78125000,86093442,91125000,171532242,177058562,226759808,233971712,617831552,664301250,686128968
%N A064205 Numbers k such that sigma(k) + tau(k) is a prime.
%C A064205 The terms involve powers of small primes. - _Jud McCranie_, Nov 29 2001
%C A064205 From _Kevin P. Thompson_, Jun 20 2022: (Start)
%C A064205 Theorem: Terms that are greater than one must be twice a square.
%C A064205 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 A064205 Theorem: No terms are congruent to 4 or 6 (mod 10).
%C A064205 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 A064205 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 A064205 128 is a term since sigma(128) + tau(128) = 255 + 8 = 263, which is prime.
%t A064205 Do[ If[ PrimeQ[ DivisorSigma[1, n] + DivisorSigma[0, n]], Print[n]], {n, 1, 10^7}]
%o A064205 (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 A064205 (Python)
%o A064205 from itertools import count, islice
%o A064205 from sympy import isprime, divisor_sigma as s, divisor_count as t
%o A064205 def agen(): # generator of terms
%o A064205     yield 1
%o A064205     yield from (k for k in (2*i*i for i in count(1)) if isprime(s(k)+t(k)))
%o A064205 print(list(islice(agen(), 30))) # _Michael S. Branicky_, Jun 20 2022
%Y A064205 Cf. A007503 (sigma+tau), A065061, A055813.
%K A064205 nonn
%O A064205 1,2
%A A064205 _Jason Earls_, Sep 21 2001
%E A064205 More terms from _Robert G. Wilson v_, Nov 12 2001
%E A064205 More terms from _Labos Elemer_, Nov 22 2001
%E A064205 More terms from _Jud McCranie_, Nov 29 2001
%E A064205 a(28) from _Harry J. Smith_, Sep 10 2009