A055813 -id:A055813 - 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

A065061 Numbers k such that sigma(k) - tau(k) is a prime.

Original entry on oeis.org

3, 8, 162, 512, 1250, 8192, 31250, 32768, 41472, 663552, 2531250, 3748322, 5120000, 6837602, 7558272, 8000000, 15780962, 33554432, 35701250, 42762752, 45334242, 68024448, 75031250, 78125000, 91125000, 137149922, 243101250, 512000000, 907039232, 959570432

Offset: 1

Jason Earls, Nov 06 2001

nonn

From Kevin P. Thompson, Jun 20 2022: (Start)
Terms greater than 3 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) - tau(162) = 363 - 10 = 353, which is prime.

Amiram Eldar, Table of n, a(n) for n = 1..5000 (terms 1..265 from Kevin P. Thompson)

I.e., A065608(n) is prime. Cf. A064205.

Cf. A000005, A000203, A023194, A038344, A055813, A062700, A115919, A141242, A229264, A229265, A229266, A229268.

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("b065061.txt", n++, " ", m); if (n==100, return)) ) } \\ Harry J. Smith, Oct 05 2009

from itertools import count, islice
from sympy import isprime, divisor_sigma as s, divisor_count as t
def agen(): # generator of terms
    yield 3
    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

a(17)-a(28) from Harry J. Smith, Oct 05 2009

a(29)-a(30) from Kevin P. Thompson, Jun 20 2022

A229264 Primes in A065387 in the order of their appearance.

Original entry on oeis.org

2, 19, 19, 79, 103, 113, 257, 523, 509, 1151, 1279, 1193, 1579, 2273, 3061, 2389, 2693, 2843, 5003, 4831, 5119, 7411, 5693, 5623, 8623, 6323, 10139, 8933, 18401, 14957, 20411, 20479, 21191, 20123, 29683, 28211, 36833, 55021, 57203, 68743, 48761, 66533, 62423

Offset: 1

Paolo P. Lava, Sep 18 2013

nonn

			Third term of A038344 is 9 and sigma(9) + phi(9) = 13 + 6 = 19 is prime.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000005, A000010, A000203, A009087, A023194, A038344, A055813, A062700, A064205, A065387, A115919, A141242, A229265, A229266, A229267, A229268.

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

Select[Table[DivisorSigma[1,n]+EulerPhi[n],{n,30000}],PrimeQ] (* Harvey P. Dale, Apr 30 2018 *)

lista(kmax) = {my(f, s); for(k = 1, kmax, f = factor(k); s= sigma(f) + eulerphi(f); if(isprime(s), print1(s, ", ")));} \\ Amiram Eldar, Nov 19 2024

Name corrected by Amiram Eldar, Nov 19 2024

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 *)

A229265 Numbers k such that sigma(k) + tau(k) + phi(k) is a prime, where sigma(k) = A000203(k), tau(k) = A000005(k) and phi(k) = A000010(k).

Original entry on oeis.org

1, 8, 200, 512, 968, 1458, 3200, 4232, 5618, 5832, 6962, 10368, 16928, 26912, 36992, 40328, 53792, 61952, 84050, 101250, 110450, 140450, 147968, 220448, 247808, 249218, 253472, 257762, 279752, 282752, 320000, 336200, 344450, 359552, 361250, 445568, 472392, 512072

Offset: 1

Paolo P. Lava, Sep 18 2013

nonn

			sigma(200) = 465, tau(200) = 12, phi(200) = 80 and 465 + 12 + 80 = 557 is prime.

Cf. A000005, A000010, A000203, A009087, A023194, A038344, A055813, A062700, A064205, A115919, A141242, A229264, A229266-A229268.

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

A272060 Numbers k such that sigma((k-1)/2) + tau((k-1)/2) is prime.

Original entry on oeis.org

3, 5, 17, 257, 325, 1025, 65537, 82945, 202501, 250001, 2829125, 7496645, 10240001, 13675205, 16000001, 27060805, 48469445, 71402501, 133448705, 150062501, 156250001, 172186885, 182250001, 343064485, 354117125, 453519617, 467943425, 1235663105

Offset: 1

Jaroslav Krizek, Apr 19 2016

nonn

Numbers k such that A000203((k-1)/2) + A000005((k-1)/2) is a prime q.
Corresponding values of primes q are in A055813.
Prime terms are in A272061.
The first 5 known Fermat primes from A019434 are in this sequence.

			sigma((17-1)/2) + tau((17-1)/2) = sigma(8) + tau(8) = 15 + 4 = 19; 19 is prime, so 17 is in the sequence.

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

Cf. A000005, A000203, A007503, A019434, A055813, A064205, A272061.

[n: n in [3..1000000] | IsPrime(NumberOfDivisors((n-1) div 2) + SumOfDivisors((n-1) div 2)) and (n-1) mod 2 eq 0];

Select[Range[3, 10^7, 2], PrimeQ[DivisorSigma[1, #] + DivisorSigma[0, #]] &[(# - 1)/2] &] (* Michael De Vlieger, Apr 20 2016 *)

isok(n) = isprime(sigma((n-1)/2) + numdiv((n-1)/2));
lista(nn) = forstep (n=3, nn, 2, if (isok(n), print1(n, ", "))); \\ Michel Marcus, Apr 19 2016

is(n)=my(f=factor(n\2)); n>2 && isprime(sigma(f)+numdiv(f)) && isprime(n) \\ Charles R Greathouse IV, Apr 29 2016

a(n) = 2*A064205(n) + 1.

A272061 Primes p such that sigma((p-1)/2) + tau((p-1)/2) is prime.

Original entry on oeis.org

3, 5, 17, 257, 65537, 453519617, 1372257937, 1927561217, 21320672257, 76001667857, 138388464037, 1216026685697, 2085136000001, 8503056000001, 30118144000001, 35427446793217, 37015056000001, 83037656250001, 87329473560577, 97850397828097, 222330465562501, 233952748524197

Offset: 1

Jaroslav Krizek, Apr 19 2016

nonn

Primes p such that A007503((p-1)/2) is a prime q.
Corresponding values of primes q: 2, 5, 19, 263, 65551, 496922891, ...
Prime terms from A272060.
The first 5 known Fermat primes from A019434 are in this sequence.
Primes of the form 2*m+1 with m a term of A064205. - Michel Marcus, Apr 25 2016

			sigma((17-1)/2) + tau((17-1)/2) = sigma(8) + tau(8) = 15 + 4 = 19; 19 is prime, so 17 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..235

Cf. A000005, A000203, A007503, A055813, A064205, A272060.

[n: n in [3..1000000] | IsPrime(n) and IsPrime(NumberOfDivisors((n-1) div 2) + SumOfDivisors((n-1) div 2)) and (n-1) mod 2 eq 0];

with(numtheory): A272061:=n->`if`(isprime(n) and isprime(sigma((n-1)/2)+tau((n-1)/2)), n, NULL): seq(A272061(n), n=3..10^5); # Wesley Ivan Hurt, Apr 20 2016

Select[Prime[Range[10000]],PrimeQ[DivisorSigma[1,(#-1)/2] + DivisorSigma[0,(#-1)/2]] & ] (* Robert Price, Apr 21 2016 *)

isok(n) = isprime(sigma((n-1)/2) + numdiv((n-1)/2));
lista(nn) = forprime (p=3, nn, if (isok(p), print1(p, ", "))); \\ Michel Marcus, Apr 19 2016

is(n)=my(f=factor(n\2)); isprime(sigma(f)+numdiv(f)) && isprime(n) \\ Charles R Greathouse IV, Apr 28 2016

a(7)-a(8) from Michel Marcus, Apr 24 2016
a(9) from Charles R Greathouse IV, Apr 29 2016
a(10) from Charles R Greathouse IV, Apr 29 2016
a(11)-a(20), using A064205 bfile, added by Michel Marcus, Nov 23 2022
a(21)-a(22) from Amiram Eldar, Dec 06 2022

cp's OEIS Frontend

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

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A065061 Numbers k such that sigma(k) - tau(k) is a prime.

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A229264 Primes in A065387 in the order of their appearance.

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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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A229265 Numbers k such that sigma(k) + tau(k) + phi(k) is a prime, where sigma(k) = A000203(k), tau(k) = A000005(k) and phi(k) = A000010(k).

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A272060 Numbers k such that sigma((k-1)/2) + tau((k-1)/2) is prime.

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