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 A064205 Numbers k such that sigma(k) + tau(k) is a prime. 10
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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) LINKS Kevin P. Thompson, Table of n, a(n) for n = 1..276 (first 34 terms from Harry J. Smith) EXAMPLE 128 is a term since sigma(128) + tau(128) = 255 + 8 = 263, which is prime. MATHEMATICA Do[ If[ PrimeQ[ DivisorSigma[1, n] + DivisorSigma[0, n]], Print[n]], {n, 1, 10^7}] PROG (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 CROSSREFS Cf. A065061, A055813. Sequence in context: A202648 A193481 A156497 * A081856 A038533 A139290 Adjacent sequences:  A064202 A064203 A064204 * A064206 A064207 A064208 KEYWORD nonn AUTHOR Jason Earls, Sep 21 2001 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 STATUS approved

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Last modified August 12 09:31 EDT 2022. Contains 356069 sequences. (Running on oeis4.)