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A064205
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Numbers k such that sigma(k) + tau(k) is a prime.
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11
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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
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OFFSET
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1,2
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COMMENTS
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The terms involve powers of small primes. - Jud McCranie, Nov 29 2001
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)
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LINKS
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EXAMPLE
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128 is a term since sigma(128) + tau(128) = 255 + 8 = 263, which is prime.
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MATHEMATICA
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Do[ If[ PrimeQ[ DivisorSigma[1, n] + DivisorSigma[0, n]], Print[n]], {n, 1, 10^7}]
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PROG
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(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)))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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