A038344 -id:A038344 - cp's OEIS frontend

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

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

A115919 Numbers k such that sigma(k) - phi(k) is a prime number.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 29, 31, 32, 36, 37, 41, 43, 47, 50, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 225, 227

Offset: 1

Giovanni Resta, Feb 06 2006

nonn

			sigma(81) - phi(81) = 67, a prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A038344.

Select[Range[300],PrimeQ[DivisorSigma[1,#]-EulerPhi[#]]&]  (* Harvey P. Dale, Feb 25 2011 *)

is(n)=isprime(sigma(n)-eulerphi(n)) \\ Charles R Greathouse IV, Nov 27 2013

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);

A230770 Numbers n such that sigma(n) + phi(n) is a composite number of the form p^k where p is a prime.

Original entry on oeis.org

2, 4, 12, 15, 110, 121, 125, 511, 908, 2047, 31269, 58252, 180544, 2275680, 3776877, 4164717, 4835820, 8386433, 8388607, 32284479, 60333777, 82628532, 122016110, 174438012, 238609292, 513528686, 515718093, 919749786, 1043394771, 3851465145, 4264386607

Offset: 1

Jahangeer Kholdi, Jan 07 2014

All semiprimes of the form 2^m-1 are in the sequence. Because if 2^m-1=p*q where p and q are prime then sigma(2^m-1)+phi(2^m-1)=(p+1)*(q+1)+(p-1)*(q-1)=2(p*q+1)=2^(m+1). 15, 511, 2047, 8388607 and 137438953471 are the first five such terms of the sequence.
Also if p=(2^m-5)/9 is prime then n=4*p is in the sequence. Because phi(n)+sigma(n)=9*p+5=2^m. 12, 908, 58952 and 77433143050453552574776799557806810784652 are the first four such terms of the sequence.
Let h(n)=sigma(n)+phi(n), except for n=4 and n=121 for all other known terms n of the sequence h(n) is of the form 2^m. Note that h(4)=3^2 and h(121)=3^5, what is the next term n of the sequence such that h(n) is odd?

			sigma(12)+phi(12)=sigma(15)+phi(15)=2^5,
sigma(180544)+phi(180544)=2^19.

Cf. A000010, A000203, A038344, A065387.

h[n_]:=DivisorSigma[1,n]+EulerPhi[n];Do[a=h[n];If[Length[FactorInteger[a]] == 1 && !PrimeQ[a], Print[n]],{n, 123456789}]

is(n)=isprimepower(sigma(n)+eulerphi(n))>1 \\ Charles R Greathouse IV, Sep 04 2014

a(24)-a(31) from Donovan Johnson, Feb 19 2014

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

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A115919 Numbers k such that sigma(k) - phi(k) is a prime number.

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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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A230770 Numbers n such that sigma(n) + phi(n) is a composite number of the form p^k where p is a prime.

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