A115919 -id:A115919

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A065061

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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)

LINKS

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

EXAMPLE

162 is a term since sigma(162) - tau(162) = 363 - 10 = 353, 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("b065061.txt", n++, " ", m); if (n==100, return)) ) } \\ Harry J. Smith, Oct 05 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 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

CROSSREFS

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

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

KEYWORD

nonn

AUTHOR

Jason Earls, Nov 06 2001

EXTENSIONS

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

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

STATUS

approved

A229264

Prime terns in A065387 in the order of their appearance.

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

EXAMPLE

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

MAPLE

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);

MATHEMATICA

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

PROG

(PARI) 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

CROSSREFS

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

KEYWORD

nonn,changed

AUTHOR

Paolo P. Lava, Sep 18 2013

EXTENSIONS

Name corrected by Amiram Eldar, Nov 19 2024

STATUS

approved

A229265

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..38.

EXAMPLE

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

MAPLE

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);

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Paolo P. Lava, Sep 18 2013

STATUS

approved

A229266

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..36.

EXAMPLE

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

MAPLE

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);

MATHEMATICA

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Paolo P. Lava, Sep 18 2013

STATUS

approved

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