A065061 -id:A065061 - cp's OEIS frontend

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

Jason Earls, Sep 21 2001

nonn

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)

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

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)

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

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

{ 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

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

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

A229268 Primes of the form sigma(k) - tau(k), where sigma(k) = A000203(k) and tau(k) = A000005(k).

Original entry on oeis.org

2, 11, 353, 1013, 2333, 16369, 58579, 65519, 123733, 1982273, 7089683, 5778653, 12795053, 10500593, 22586027, 19980143, 24126653, 67108837, 72494713, 90781993, 106199593, 203275951, 164118923, 183105421, 320210549, 259997173, 794091653, 1279963973

Offset: 1

Paolo P. Lava, Sep 18 2013

nonn

			Second term of A065061 is 8 and sigma(8) - tau(8) = 15 - 4 = 11 is prime.

Amiram Eldar, Table of n, a(n) for n = 1..5000

Cf. A000005, A000010, A000203, A009087, A023194, A038344, A055813, A062700, A064205, A065608, A141242, A229264, A229266

with(numtheory); P:=proc(q) local a,n; a:= sigma(n)-tau(n); for n from 1 to q do
if isprime(a) then print(a); fi; od; end: P(10^6);

Join[{2}, Select[(DivisorSigma[1, #] - DivisorSigma[0, #]) & /@ (2*Range[20000]^2), PrimeQ]] (* Amiram Eldar, Dec 06 2022 *)

a(n) = A000203(A065061(n)) - A000005(A065061(n)). - Michel Marcus, Sep 21 2013

a(n) = A065608(A065061(n)). - Amiram Eldar, Dec 06 2022

More terms from Michel Marcus, Sep 21 2013

A229266 Primes of the form sigma(k) + tau(k) + phi(k), where sigma(k) = A000203(k), tau(k) = A000005(k) and phi(k) = A000010(k).

Original entry on oeis.org

3, 23, 557, 1289, 2447, 3779, 9209, 10331, 11351, 18367, 14051, 34351, 42953, 67883, 95717, 96587, 134807, 164249, 193057, 310553, 253159, 321397, 383723, 548213, 657311, 499151, 630023, 516251, 732181, 713927, 927013, 932431, 784627, 906473, 855331, 1121987

Offset: 1

Paolo P. Lava, Sep 18 2013

nonn

			The third term of A229265 is 200 and sigma(200) +  tau(200) + phi(200) = 465 + 12 + 80 = 557 is prime.

Cf. A000005, A000010, A000203, A009087, A023194, A038344, A055813, A062700, A064205, A115919, A141242, A229264, A229265, A065061, A229268.

with(numtheory); P:=proc(q) local a, n; for n from 1 to q do a:=sigma(n)+tau(n)+phi(n);
if isprime(a) then print(a); fi; od; end: P(10^6);

Select[Table[DivisorSigma[0,n]+DivisorSigma[1,n]+EulerPhi[n],{n,10^6}],PrimeQ] (* Harvey P. Dale, Oct 03 2023 *)

A065116 Numbers k such that sigma(k) + tau(k) and sigma(k) - tau(k) are primes.

Original entry on oeis.org

8, 162, 512, 32768, 41472, 3748322, 5120000, 6837602, 8000000, 35701250, 75031250, 78125000, 91125000, 907039232, 10660336128, 11911961250, 21234895362, 41265473762, 55965865922, 209642370242, 835707290112, 1148179179938, 1821331173888, 2097152000000

Offset: 1

Robert G. Wilson v, Nov 12 2001

nonn

Intersection of A064205 and A065061.
From Kevin P. Thompson, Jun 20 2022: (Start)
Terms must be twice a square (see A064205).
No terms are congruent to 4 or 6 (mod 10) (see A064205). (End)

			162 is a term since sigma(162) = 363 and tau(162) = 10 are numbers whose sum (373) and difference (353) are both primes.

Donovan Johnson, Table of n, a(n) for n = 1..250

Cf. A064205, A065061.

Do[ds1 = DivisorSigma[1, n]; ds0 = DivisorSigma[0, n]; If[ PrimeQ[ds1 + ds0] && PrimeQ[ds1 - ds0], Print[n]], {n, 1, 10^7} ]

a(10)-a(19) from Donovan Johnson, Jul 09 2010

a(20)-a(24) from Donovan Johnson, Aug 23 2013

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

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A229268 Primes of the form sigma(k) - tau(k), where sigma(k) = A000203(k) and tau(k) = A000005(k).

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A229266 Primes of the form sigma(k) + tau(k) + phi(k), where sigma(k) = A000203(k), tau(k) = A000005(k) and phi(k) = A000010(k).

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A065116 Numbers k such that sigma(k) + tau(k) and sigma(k) - tau(k) are primes.

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